/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup,lookup,run} and constructors {Cons,Nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup,lookup,run} and constructors {Cons,Nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup,lookup,run} and constructors {Cons,Nil} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "Cons") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "Nil") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "intlookup") :: ["A"(0, 0, 0) x "A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) F (TrsFun "lookup") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "main") :: ["A"(0, 0, 0) x "A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) F (TrsFun "run") :: ["A"(0, 0, 0) x "A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) main(x1,x2) -> run(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup,lookup,run} and constructors {Cons,Nil} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: lookup(y,u){y -> Cons(x,y),u -> Cons(z,u)} = lookup(Cons(x,y),Cons(z,u)) ->^+ lookup(y,u) = C[lookup(y,u) = lookup(y,u){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup,lookup,run} and constructors {Cons,Nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) lookup#(Nil(),Cons(x,xs)) -> c_3() run#(e,p) -> c_4(intlookup#(e,p)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) lookup#(Nil(),Cons(x,xs)) -> c_3() run#(e,p) -> c_4(intlookup#(e,p)) - Weak TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {1,2}. Here rules are labelled as follows: 1: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) 2: lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) 3: lookup#(Nil(),Cons(x,xs)) -> c_3() 4: run#(e,p) -> c_4(intlookup#(e,p)) ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) run#(e,p) -> c_4(intlookup#(e,p)) - Weak DPs: lookup#(Nil(),Cons(x,xs)) -> c_3() - Weak TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) -->_2 lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)):2 -->_2 lookup#(Nil(),Cons(x,xs)) -> c_3():4 -->_1 intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)):1 2:S:lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) -->_1 lookup#(Nil(),Cons(x,xs)) -> c_3():4 -->_1 lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)):2 3:S:run#(e,p) -> c_4(intlookup#(e,p)) -->_1 intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)):1 4:W:lookup#(Nil(),Cons(x,xs)) -> c_3() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: lookup#(Nil(),Cons(x,xs)) -> c_3() ** Step 1.b:4: RemoveHeads. MAYBE + Considered Problem: - Strict DPs: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) run#(e,p) -> c_4(intlookup#(e,p)) - Weak TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) -->_2 lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)):2 -->_1 intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)):1 2:S:lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) -->_1 lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)):2 3:S:run#(e,p) -> c_4(intlookup#(e,p)) -->_1 intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)):1 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). [(3,run#(e,p) -> c_4(intlookup#(e,p)))] ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) - Weak TRS: intlookup(e,p) -> intlookup(lookup(e,p),p) lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x run(e,p) -> intlookup(e,p) - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) - Weak TRS: lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) and a lower component lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) Further, following extension rules are added to the lower component. intlookup#(e,p) -> intlookup#(lookup(e,p),p) intlookup#(e,p) -> lookup#(e,p) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) - Weak TRS: lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)) -->_1 intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p),lookup#(e,p)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p)) *** Step 1.b:6.a:2: Failure MAYBE + Considered Problem: - Strict DPs: intlookup#(e,p) -> c_1(intlookup#(lookup(e,p),p)) - Weak TRS: lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) - Weak DPs: intlookup#(e,p) -> intlookup#(lookup(e,p),p) intlookup#(e,p) -> lookup#(e,p) - Weak TRS: lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + 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_2) = {1} Following symbols are considered usable: {intlookup#,lookup#,run#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [4] p(Nil) = [4] p(intlookup) = [1] x1 + [1] p(lookup) = [4] x1 + [6] p(run) = [8] x2 + [2] p(intlookup#) = [4] x2 + [12] p(lookup#) = [4] x2 + [0] p(run#) = [1] x1 + [2] p(c_1) = [1] x2 + [1] p(c_2) = [1] x1 + [14] p(c_3) = [2] p(c_4) = [4] Following rules are strictly oriented: lookup#(Cons(x',xs'),Cons(x,xs)) = [4] xs + [16] > [4] xs + [14] = c_2(lookup#(xs',xs)) Following rules are (at-least) weakly oriented: intlookup#(e,p) = [4] p + [12] >= [4] p + [12] = intlookup#(lookup(e,p),p) intlookup#(e,p) = [4] p + [12] >= [4] p + [0] = lookup#(e,p) *** Step 1.b:6.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: intlookup#(e,p) -> intlookup#(lookup(e,p),p) intlookup#(e,p) -> lookup#(e,p) lookup#(Cons(x',xs'),Cons(x,xs)) -> c_2(lookup#(xs',xs)) - Weak TRS: lookup(Cons(x',xs'),Cons(x,xs)) -> lookup(xs',xs) lookup(Nil(),Cons(x,xs)) -> x - Signature: {intlookup/2,lookup/2,run/2,intlookup#/2,lookup#/2,run#/2} / {Cons/2,Nil/0,c_1/2,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {intlookup#,lookup#,run#} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),?)