WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: deleteMin#1(E()) -> ErrorHeap() deleteMin#1(T(E(),x6,x8)) -> x8 deleteMin#1(T(T(E(),x24,x28),x12,x16)) -> T(x28,x12,x16) deleteMin#1(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> T(deleteMin#1(T(x32,x36,x40)),x24,T(x28,x12,x16)) main(x0) -> deleteMin#1(x0) - Signature: {deleteMin#1/1,main/1} / {E/0,ErrorHeap/0,T/3} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1,main} and constructors {E,ErrorHeap,T} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs deleteMin#1#(E()) -> c_1() deleteMin#1#(T(E(),x6,x8)) -> c_2() deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) main#(x0) -> c_5(deleteMin#1#(x0)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: deleteMin#1#(E()) -> c_1() deleteMin#1#(T(E(),x6,x8)) -> c_2() deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) main#(x0) -> c_5(deleteMin#1#(x0)) - Strict TRS: deleteMin#1(E()) -> ErrorHeap() deleteMin#1(T(E(),x6,x8)) -> x8 deleteMin#1(T(T(E(),x24,x28),x12,x16)) -> T(x28,x12,x16) deleteMin#1(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> T(deleteMin#1(T(x32,x36,x40)),x24,T(x28,x12,x16)) main(x0) -> deleteMin#1(x0) - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: deleteMin#1#(E()) -> c_1() deleteMin#1#(T(E(),x6,x8)) -> c_2() deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) main#(x0) -> c_5(deleteMin#1#(x0)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: deleteMin#1#(E()) -> c_1() deleteMin#1#(T(E(),x6,x8)) -> c_2() deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) main#(x0) -> c_5(deleteMin#1#(x0)) - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3} by application of Pre({1,2,3}) = {4,5}. Here rules are labelled as follows: 1: deleteMin#1#(E()) -> c_1() 2: deleteMin#1#(T(E(),x6,x8)) -> c_2() 3: deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() 4: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) 5: main#(x0) -> c_5(deleteMin#1#(x0)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) main#(x0) -> c_5(deleteMin#1#(x0)) - Weak DPs: deleteMin#1#(E()) -> c_1() deleteMin#1#(T(E(),x6,x8)) -> c_2() deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) -->_1 deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3():5 -->_1 deleteMin#1#(T(E(),x6,x8)) -> c_2():4 -->_1 deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))):1 2:S:main#(x0) -> c_5(deleteMin#1#(x0)) -->_1 deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3():5 -->_1 deleteMin#1#(T(E(),x6,x8)) -> c_2():4 -->_1 deleteMin#1#(E()) -> c_1():3 -->_1 deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))):1 3:W:deleteMin#1#(E()) -> c_1() 4:W:deleteMin#1#(T(E(),x6,x8)) -> c_2() 5:W:deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: deleteMin#1#(E()) -> c_1() 4: deleteMin#1#(T(E(),x6,x8)) -> c_2() 5: deleteMin#1#(T(T(E(),x24,x28),x12,x16)) -> c_3() * Step 5: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) main#(x0) -> c_5(deleteMin#1#(x0)) - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) -->_1 deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))):1 2:S:main#(x0) -> c_5(deleteMin#1#(x0)) -->_1 deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(2,main#(x0) -> c_5(deleteMin#1#(x0)))] * Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + 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: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) The strictly oriented rules are moved into the weak component. ** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + 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_4) = {1} Following symbols are considered usable: {deleteMin#1#,main#} TcT has computed the following interpretation: p(E) = [4] p(ErrorHeap) = [1] p(T) = [1] x1 + [8] p(deleteMin#1) = [1] x1 + [1] p(main) = [1] x1 + [0] p(deleteMin#1#) = [1] x1 + [0] p(main#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [8] p(c_4) = [1] x1 + [14] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) = [1] x32 + [24] > [1] x32 + [22] = c_4(deleteMin#1#(T(x32,x36,x40))) Following rules are (at-least) weakly oriented: ** Step 6.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + 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: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) -->_1 deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: deleteMin#1#(T(T(T(x32,x36,x40),x24,x28),x12,x16)) -> c_4(deleteMin#1#(T(x32,x36,x40))) ** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {deleteMin#1/1,main/1,deleteMin#1#/1,main#/1} / {E/0,ErrorHeap/0,T/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {deleteMin#1#,main#} and constructors {E,ErrorHeap,T} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))