WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,0()) -> x -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) -> 0() p(0()) -> 0() p(s(x)) -> x - Signature: {-/2,p/1} / {0/0,greater/2,if/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs -#(x,0()) -> c_1() -#(x,s(y)) -> c_2(-#(x,p(s(y)))) -#(0(),y) -> c_3() p#(0()) -> c_4() p#(s(x)) -> c_5() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(x,0()) -> c_1() -#(x,s(y)) -> c_2(-#(x,p(s(y)))) -#(0(),y) -> c_3() p#(0()) -> c_4() p#(s(x)) -> c_5() - Strict TRS: -(x,0()) -> x -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) -> 0() p(0()) -> 0() p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: p(s(x)) -> x -#(x,0()) -> c_1() -#(x,s(y)) -> c_2(-#(x,p(s(y)))) -#(0(),y) -> c_3() p#(0()) -> c_4() p#(s(x)) -> c_5() * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(x,0()) -> c_1() -#(x,s(y)) -> c_2(-#(x,p(s(y)))) -#(0(),y) -> c_3() p#(0()) -> c_4() p#(s(x)) -> c_5() - Strict TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,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(-#) = {2}, uargs(c_2) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [0] p(0) = [0] p(greater) = [1] x1 + [1] x2 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [11] p(-#) = [1] x2 + [0] p(p#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] Following rules are strictly oriented: p(s(x)) = [1] x + [11] > [1] x + [0] = x Following rules are (at-least) weakly oriented: -#(x,0()) = [0] >= [0] = c_1() -#(x,s(y)) = [1] y + [11] >= [1] y + [11] = c_2(-#(x,p(s(y)))) -#(0(),y) = [1] y + [0] >= [0] = c_3() p#(0()) = [0] >= [0] = c_4() p#(s(x)) = [0] >= [0] = c_5() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(x,0()) -> c_1() -#(x,s(y)) -> c_2(-#(x,p(s(y)))) -#(0(),y) -> c_3() p#(0()) -> c_4() p#(s(x)) -> c_5() - Weak TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,5} by application of Pre({1,3,4,5}) = {2}. Here rules are labelled as follows: 1: -#(x,0()) -> c_1() 2: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) 3: -#(0(),y) -> c_3() 4: p#(0()) -> c_4() 5: p#(s(x)) -> c_5() * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) - Weak DPs: -#(x,0()) -> c_1() -#(0(),y) -> c_3() p#(0()) -> c_4() p#(s(x)) -> c_5() - Weak TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:-#(x,s(y)) -> c_2(-#(x,p(s(y)))) -->_1 -#(0(),y) -> c_3():3 -->_1 -#(x,0()) -> c_1():2 -->_1 -#(x,s(y)) -> c_2(-#(x,p(s(y)))):1 2:W:-#(x,0()) -> c_1() 3:W:-#(0(),y) -> c_3() 4:W:p#(0()) -> c_4() 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. 5: p#(s(x)) -> c_5() 4: p#(0()) -> c_4() 2: -#(x,0()) -> c_1() 3: -#(0(),y) -> c_3() * Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) - Weak TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) The strictly oriented rules are moved into the weak component. ** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) - Weak TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1} Following symbols are considered usable: {p,-#,p#} TcT has computed the following interpretation: p(-) = [4 0 4] [0 0 0] [0] [0 2 1] x1 + [0 0 0] x2 + [0] [0 0 2] [1 4 4] [0] p(0) = [0] [1] [0] p(greater) = [0 0 1] [0 1 1] [0] [0 0 2] x1 + [0 0 2] x2 + [1] [0 0 0] [0 0 0] [1] p(if) = [1 1 0] [0 0 0] [0 1 1] [1] [0 0 4] x1 + [0 0 4] x2 + [0 0 2] x3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] p(p) = [4 1 4] [0] [2 4 0] x1 + [1] [0 1 0] [0] p(s) = [1 2 0] [0] [0 0 1] x1 + [2] [0 0 1] [3] p(-#) = [0 1 0] [0 0 4] [1] [4 4 0] x1 + [1 0 0] x2 + [0] [0 0 1] [1 0 0] [0] p(p#) = [0] [1] [0] p(c_1) = [0] [0] [2] p(c_2) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(c_3) = [1] [0] [2] p(c_4) = [0] [1] [1] p(c_5) = [1] [0] [1] Following rules are strictly oriented: -#(x,s(y)) = [0 1 0] [0 0 4] [13] [4 4 0] x + [1 2 0] y + [0] [0 0 1] [1 2 0] [0] > [0 1 0] [0 0 4] [10] [0 0 0] x + [0 0 0] y + [0] [0 0 0] [0 0 0] [0] = c_2(-#(x,p(s(y)))) Following rules are (at-least) weakly oriented: p(s(x)) = [4 8 5] [14] [2 4 4] x + [9] [0 0 1] [2] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x ** Step 6.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) - Weak TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,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: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) - Weak TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:-#(x,s(y)) -> c_2(-#(x,p(s(y)))) -->_1 -#(x,s(y)) -> c_2(-#(x,p(s(y)))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: -#(x,s(y)) -> c_2(-#(x,p(s(y)))) ** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: p(s(x)) -> x - Signature: {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))