WORST_CASE(?,O(n^1)) * Step 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: innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost 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() minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5() Weak DPs and mark the set of starting terms. * Step 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() minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5() - 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_3/0,c_4/1,c_5/0} - Obligation: innermost 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() minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5() * Step 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() minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5() - 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_3/0,c_4/1,c_5/0} - Obligation: innermost 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) = [0] p(div) = [2] x1 + [2] x2 + [0] p(minus) = [1] x1 + [13] p(p) = [1] x1 + [0] p(s) = [1] x1 + [8] p(div#) = [1] x1 + [2] x2 + [0] p(minus#) = [1] x1 + [0] p(p#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] Following rules are strictly oriented: div#(0(),s(Y)) = [2] Y + [16] > [0] = c_1() p#(s(X)) = [1] X + [8] > [0] = c_5() minus(X,0()) = [1] X + [13] > [1] X + [0] = X minus(s(X),s(Y)) = [1] X + [21] > [1] X + [13] = p(minus(X,Y)) p(s(X)) = [1] X + [8] > [1] X + [0] = X Following rules are (at-least) weakly oriented: div#(s(X),s(Y)) = [1] X + [2] Y + [24] >= [1] X + [2] Y + [29] = c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) = [1] X + [0] >= [0] = c_3() minus#(s(X),s(Y)) = [1] X + [8] >= [1] X + [13] = c_4(p#(minus(X,Y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) minus#(X,0()) -> c_3() minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) - Weak DPs: div#(0(),s(Y)) -> c_1() p#(s(X)) -> c_5() - 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_3/0,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3} by application of Pre({2,3}) = {}. Here rules are labelled as follows: 1: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) 2: minus#(X,0()) -> c_3() 3: minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) 4: div#(0(),s(Y)) -> c_1() 5: p#(s(X)) -> c_5() * Step 5: 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() minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) p#(s(X)) -> c_5() - 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_3/0,c_4/1,c_5/0} - Obligation: innermost 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() 4:W:minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) -->_1 p#(s(X)) -> c_5():5 5:W:p#(s(X)) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y))) 5: p#(s(X)) -> c_5() 3: minus#(X,0()) -> c_3() 2: div#(0(),s(Y)) -> c_1() * Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - 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_3/0,c_4/1,c_5/0} - Obligation: innermost 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 6.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 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_3/0,c_4/1,c_5/0} - Obligation: innermost 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} Following symbols are considered usable: {minus,p,div#,minus#,p#} TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [1] x2 + [0] p(minus) = [1] x1 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [1] p(div#) = [4] x1 + [9] p(minus#) = [2] x2 + [0] p(p#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [2] p(c_3) = [4] p(c_4) = [1] x1 + [0] p(c_5) = [0] Following rules are strictly oriented: div#(s(X),s(Y)) = [4] X + [13] > [4] X + [11] = c_2(div#(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: minus(X,0()) = [1] X + [0] >= [1] X + [0] = X minus(s(X),s(Y)) = [1] X + [1] >= [1] X + [0] = p(minus(X,Y)) p(s(X)) = [1] X + [1] >= [1] X + [0] = X ** Step 6.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - 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_3/0,c_4/1,c_5/0} - Obligation: innermost 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}} + Details: () ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - 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_3/0,c_4/1,c_5/0} - Obligation: innermost 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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) ** Step 6.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_3/0,c_4/1,c_5/0} - Obligation: innermost 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(?,O(n^1))