/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,pred,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: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,pred,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: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,pred,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"(0) F (TrsFun "0") :: [] -(0)-> "A"(1) F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "minus") :: ["A"(1) x "A"(0)] -(1)-> "A"(1) F (TrsFun "pred") :: ["A"(1)] -(0)-> "A"(1) F (TrsFun "quot") :: ["A"(1) 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: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) main(x1,x2) -> quot(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,pred,quot} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: minus(x,y){y -> s(y)} = minus(x,s(y)) ->^+ pred(minus(x,y)) = C[minus(x,y) = minus(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,pred,quot} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs minus#(x,0()) -> c_1() minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)) pred#(s(x)) -> c_3() quot#(0(),s(y)) -> c_4() quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: minus#(x,0()) -> c_1() minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)) pred#(s(x)) -> c_3() quot#(0(),s(y)) -> c_4() quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4} by application of Pre({1,3,4}) = {2,5}. Here rules are labelled as follows: 1: minus#(x,0()) -> c_1() 2: minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)) 3: pred#(s(x)) -> c_3() 4: quot#(0(),s(y)) -> c_4() 5: quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)) quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: minus#(x,0()) -> c_1() pred#(s(x)) -> c_3() quot#(0(),s(y)) -> c_4() - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)) -->_1 pred#(s(x)) -> c_3():4 -->_2 minus#(x,0()) -> c_1():3 -->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1 2:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(0(),s(y)) -> c_4():5 -->_2 minus#(x,0()) -> c_1():3 -->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2 -->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1 3:W:minus#(x,0()) -> c_1() 4:W:pred#(s(x)) -> c_3() 5:W:quot#(0(),s(y)) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: quot#(0(),s(y)) -> c_4() 3: minus#(x,0()) -> c_1() 4: pred#(s(x)) -> c_3() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)) quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)) -->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1 2:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2 -->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(x,s(y)) -> c_2(minus#(x,y)) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: minus#(x,s(y)) -> c_2(minus#(x,y)) quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x minus#(x,s(y)) -> c_2(minus#(x,y)) quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) ** Step 1.b:6: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: minus#(x,s(y)) -> c_2(minus#(x,y)) quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + 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 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) and a lower component minus#(x,s(y)) -> c_2(minus#(x,y)) Further, following extension rules are added to the lower component. quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) *** Step 1.b:6.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y))) *** Step 1.b:6.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(pred) = {1}, uargs(quot#) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(minus) = [1] x1 + [2] p(pred) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [3] p(minus#) = [0] p(pred#) = [2] x1 + [1] p(quot#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: quot#(s(x),s(y)) = [1] x + [3] > [1] x + [2] = c_5(quot#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: minus(x,0()) = [1] x + [2] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [2] >= [1] x + [2] = pred(minus(x,y)) pred(s(x)) = [1] x + [3] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(x,s(y)) -> c_2(minus#(x,y)) - Weak DPs: quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,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_2) = {1} Following symbols are considered usable: {minus#,pred#,quot#} TcT has computed the following interpretation: p(0) = [0] p(minus) = [0] p(pred) = [1] x1 + [0] p(quot) = [8] x2 + [0] p(s) = [1] x1 + [2] p(minus#) = [8] x2 + [2] p(pred#) = [1] x1 + [8] p(quot#) = [12] x2 + [6] p(c_1) = [0] p(c_2) = [1] x1 + [14] p(c_3) = [2] p(c_4) = [1] p(c_5) = [2] x2 + [0] Following rules are strictly oriented: minus#(x,s(y)) = [8] y + [18] > [8] y + [16] = c_2(minus#(x,y)) Following rules are (at-least) weakly oriented: quot#(s(x),s(y)) = [12] y + [30] >= [8] y + [2] = minus#(x,y) quot#(s(x),s(y)) = [12] y + [30] >= [12] y + [30] = quot#(minus(x,y),s(y)) *** Step 1.b:6.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: minus#(x,s(y)) -> c_2(minus#(x,y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {minus#,pred#,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^2))