/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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(0(),s(Y)) -> 0() div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {div,minus,p} 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(0(),s(Y)) -> 0() div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: minus(x,y){x -> s(x),y -> s(y)} = minus(s(x),s(y)) ->^+ p(minus(x,y)) = C[minus(x,y) = minus(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak dependency pairs: Strict DPs div#(0(),s(Y)) -> c_1() div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(0(),s(Y)) -> c_1() div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X div#(0(),s(Y)) -> c_1() div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) ** Step 1.b:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(0(),s(Y)) -> c_1() div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) - Strict TRS: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + 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(p) = {1}, uargs(div#) = {1}, uargs(p#) = {1}, uargs(c_2) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(div) = [2] x2 + [0] p(minus) = [1] x1 + [4] p(p) = [1] x1 + [0] p(s) = [1] x1 + [1] p(div#) = [1] x1 + [1] x2 + [0] p(minus#) = [7] x1 + [3] x2 + [5] p(p#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [7] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: div#(0(),s(Y)) = [1] Y + [2] > [0] = c_1() minus#(X,0()) = [7] X + [8] > [7] X + [0] = c_3(X) minus#(s(X),s(Y)) = [7] X + [3] Y + [15] > [1] X + [4] = c_4(p#(minus(X,Y))) p#(s(X)) = [1] X + [1] > [1] X + [0] = c_5(X) minus(X,0()) = [1] X + [4] > [1] X + [0] = X minus(s(X),s(Y)) = [1] X + [5] > [1] X + [4] = p(minus(X,Y)) p(s(X)) = [1] X + [1] > [1] X + [0] = X Following rules are (at-least) weakly oriented: div#(s(X),s(Y)) = [1] X + [1] Y + [2] >= [1] X + [1] Y + [5] = c_2(div#(minus(X,Y),s(Y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:4: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - Weak DPs: div#(0(),s(Y)) -> c_1() minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) - Weak TRS: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) -->_1 div#(0(),s(Y)) -> c_1():2 -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))):1 2:W:div#(0(),s(Y)) -> c_1() 3:W:minus#(X,0()) -> c_3(X) -->_1 p#(s(X)) -> c_5(X):5 -->_1 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))):4 -->_1 minus#(X,0()) -> c_3(X):3 -->_1 div#(0(),s(Y)) -> c_1():2 -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))):1 4:W:minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) -->_1 p#(s(X)) -> c_5(X):5 5:W:p#(s(X)) -> c_5(X) -->_1 p#(s(X)) -> c_5(X):5 -->_1 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))):4 -->_1 minus#(X,0()) -> c_3(X):3 -->_1 div#(0(),s(Y)) -> c_1():2 -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: div#(0(),s(Y)) -> c_1() ** Step 1.b:5: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - Weak DPs: minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) - Weak TRS: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} 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: 1: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) The strictly oriented rules are moved into the weak component. *** Step 1.b:5.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - Weak DPs: minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) - Weak TRS: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} 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_2) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(div) = [4] x1 + [4] x2 + [1] p(minus) = [1] x1 + [0] p(p) = [1] x1 + [2] p(s) = [1] x1 + [2] p(div#) = [8] x1 + [0] p(minus#) = [8] x1 + [1] x2 + [0] p(p#) = [2] p(c_1) = [2] p(c_2) = [1] x1 + [13] p(c_3) = [2] p(c_4) = [4] x1 + [0] p(c_5) = [1] Following rules are strictly oriented: div#(s(X),s(Y)) = [8] X + [16] > [8] X + [13] = c_2(div#(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: minus#(X,0()) = [8] X + [2] >= [2] = c_3(X) minus#(s(X),s(Y)) = [8] X + [1] Y + [18] >= [8] = c_4(p#(minus(X,Y))) p#(s(X)) = [2] >= [1] = c_5(X) minus(X,0()) = [1] X + [0] >= [1] X + [0] = X minus(s(X),s(Y)) = [1] X + [2] >= [1] X + [2] = p(minus(X,Y)) p(s(X)) = [1] X + [4] >= [1] X + [0] = X *** Step 1.b:5.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) - Weak TRS: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} 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#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) -> c_3(X) minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5(X) - Weak TRS: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))):1 2:W:minus#(X,0()) -> c_3(X) -->_1 p#(s(X)) -> c_5(X):4 -->_1 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))):3 -->_1 minus#(X,0()) -> c_3(X):2 -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))):1 3:W:minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) -->_1 p#(s(X)) -> c_5(X):4 4:W:p#(s(X)) -> c_5(X) -->_1 p#(s(X)) -> c_5(X):4 -->_1 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))):3 -->_1 minus#(X,0()) -> c_3(X):2 -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: minus#(X,0()) -> c_3(X) 4: p#(s(X)) -> c_5(X) 3: minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) 1: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) *** Step 1.b:5.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1} - Obligation: runtime complexity wrt. defined symbols {div#,minus#,p#} 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))