/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),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: div(x,y) -> quot(x,y,y) div(0(),y) -> 0() quot(x,0(),s(z)) -> s(div(x,s(z))) quot(0(),s(y),z) -> 0() quot(s(x),s(y),z) -> quot(x,y,z) - Signature: {div/2,quot/3} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,quot} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(x,y) -> quot(x,y,y) div(0(),y) -> 0() quot(x,0(),s(z)) -> s(div(x,s(z))) quot(0(),s(y),z) -> 0() quot(s(x),s(y),z) -> quot(x,y,z) - Signature: {div/2,quot/3} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,quot} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: div(x,y) -> quot(x,y,y) div(0(),y) -> 0() quot(x,0(),s(z)) -> s(div(x,s(z))) quot(0(),s(y),z) -> 0() quot(s(x),s(y),z) -> quot(x,y,z) - Signature: {div/2,quot/3} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,quot} and constructors {0,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"(1) F (TrsFun "0") :: [] -(0)-> "A"(0) F (TrsFun "div") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "main") :: ["A"(1) x "A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "quot") :: ["A"(1) x "A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: div(x,y) -> quot(x,y,y) div(0(),y) -> 0() quot(x,0(),s(z)) -> s(div(x,s(z))) quot(0(),s(y),z) -> 0() quot(s(x),s(y),z) -> quot(x,y,z) main(x1,x2,x3) -> quot(x1,x2,x3) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(x,y) -> quot(x,y,y) div(0(),y) -> 0() quot(x,0(),s(z)) -> s(div(x,s(z))) quot(0(),s(y),z) -> 0() quot(s(x),s(y),z) -> quot(x,y,z) - Signature: {div/2,quot/3} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,quot} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: quot(x,y,z){x -> s(x),y -> s(y)} = quot(s(x),s(y),z) ->^+ quot(x,y,z) = C[quot(x,y,z) = quot(x,y,z){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(x,y) -> quot(x,y,y) div(0(),y) -> 0() quot(x,0(),s(z)) -> s(div(x,s(z))) quot(0(),s(y),z) -> 0() quot(s(x),s(y),z) -> quot(x,y,z) - Signature: {div/2,quot/3} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,quot} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs div#(x,y) -> c_1(quot#(x,y,y)) div#(0(),y) -> c_2() quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(0(),s(y),z) -> c_4() quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(x,y) -> c_1(quot#(x,y,y)) div#(0(),y) -> c_2() quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(0(),s(y),z) -> c_4() quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) - Strict TRS: div(x,y) -> quot(x,y,y) div(0(),y) -> 0() quot(x,0(),s(z)) -> s(div(x,s(z))) quot(0(),s(y),z) -> 0() quot(s(x),s(y),z) -> quot(x,y,z) - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: div#(x,y) -> c_1(quot#(x,y,y)) div#(0(),y) -> c_2() quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(0(),s(y),z) -> c_4() quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(x,y) -> c_1(quot#(x,y,y)) div#(0(),y) -> c_2() quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(0(),s(y),z) -> c_4() quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4} by application of Pre({2,4}) = {1,3,5}. Here rules are labelled as follows: 1: div#(x,y) -> c_1(quot#(x,y,y)) 2: div#(0(),y) -> c_2() 3: quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) 4: quot#(0(),s(y),z) -> c_4() 5: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) ** Step 1.b:4: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(x,y) -> c_1(quot#(x,y,y)) quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) - Weak DPs: div#(0(),y) -> c_2() quot#(0(),s(y),z) -> c_4() - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:div#(x,y) -> c_1(quot#(x,y,y)) -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3 -->_1 quot#(0(),s(y),z) -> c_4():5 2:S:quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) -->_1 div#(0(),y) -> c_2():4 -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1 3:S:quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) -->_1 quot#(0(),s(y),z) -> c_4():5 -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3 -->_1 quot#(x,0(),s(z)) -> c_3(div#(x,s(z))):2 4:W:div#(0(),y) -> c_2() 5:W:quot#(0(),s(y),z) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: div#(0(),y) -> c_2() 5: quot#(0(),s(y),z) -> c_4() ** Step 1.b:5: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(x,y) -> c_1(quot#(x,y,y)) quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,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}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) Consider the set of all dependency pairs 1: div#(x,y) -> c_1(quot#(x,y,y)) 2: quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) 3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {3} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 1.b:5.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(x,y) -> c_1(quot#(x,y,y)) quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {div#,quot#} TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [1] p(quot) = [1] x1 + [1] p(s) = [1] x1 + [8] p(div#) = [2] x1 + [0] p(quot#) = [2] x1 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [8] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [12] Following rules are strictly oriented: quot#(s(x),s(y),z) = [2] x + [16] > [2] x + [12] = c_5(quot#(x,y,z)) Following rules are (at-least) weakly oriented: div#(x,y) = [2] x + [0] >= [2] x + [0] = c_1(quot#(x,y,y)) quot#(x,0(),s(z)) = [2] x + [0] >= [2] x + [0] = c_3(div#(x,s(z))) *** Step 1.b:5.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: div#(x,y) -> c_1(quot#(x,y,y)) quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) - Weak DPs: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () *** Step 1.b:5.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(x,y) -> c_1(quot#(x,y,y)) quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:div#(x,y) -> c_1(quot#(x,y,y)) -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3 2:W:quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1 3:W:quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3 -->_1 quot#(x,0(),s(z)) -> c_3(div#(x,s(z))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: div#(x,y) -> c_1(quot#(x,y,y)) 2: quot#(x,0(),s(z)) -> c_3(div#(x,s(z))) 3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)) *** Step 1.b:5.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))