* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,constant,t} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,constant,t} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: D(x){x -> *(x,y)} = D(*(x,y)) ->^+ +(*(y,D(x)),*(x,D(y))) = C[D(x) = D(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,constant,t} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(constant()) -> c_4() D#(t()) -> c_5() Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(constant()) -> c_4() D#(t()) -> c_5() - Strict TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(t()) -> 1() - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(constant()) -> c_4() D#(t()) -> c_5() ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(constant()) -> c_4() D#(t()) -> c_5() - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,5} by application of Pre({4,5}) = {1,2,3}. Here rules are labelled as follows: 1: D#(*(x,y)) -> c_1(D#(x),D#(y)) 2: D#(+(x,y)) -> c_2(D#(x),D#(y)) 3: D#(-(x,y)) -> c_3(D#(x),D#(y)) 4: D#(constant()) -> c_4() 5: D#(t()) -> c_5() ** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Weak DPs: D#(constant()) -> c_4() D#(t()) -> c_5() - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:D#(*(x,y)) -> c_1(D#(x),D#(y)) -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(t()) -> c_5():5 -->_1 D#(t()) -> c_5():5 -->_2 D#(constant()) -> c_4():4 -->_1 D#(constant()) -> c_4():4 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 2:S:D#(+(x,y)) -> c_2(D#(x),D#(y)) -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(t()) -> c_5():5 -->_1 D#(t()) -> c_5():5 -->_2 D#(constant()) -> c_4():4 -->_1 D#(constant()) -> c_4():4 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 3:S:D#(-(x,y)) -> c_3(D#(x),D#(y)) -->_2 D#(t()) -> c_5():5 -->_1 D#(t()) -> c_5():5 -->_2 D#(constant()) -> c_4():4 -->_1 D#(constant()) -> c_4():4 -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 4:W:D#(constant()) -> c_4() 5:W:D#(t()) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: D#(constant()) -> c_4() 5: D#(t()) -> c_5() ** Step 1.b:5: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,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: 3: D#(-(x,y)) -> c_3(D#(x),D#(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: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,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_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2} Following symbols are considered usable: {D#} TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [0] p(+) = [1] x1 + [1] x2 + [0] p(-) = [1] x1 + [1] x2 + [1] p(0) = [0] p(1) = [1] p(D) = [0] p(constant) = [2] p(t) = [0] p(D#) = [8] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [4] p(c_4) = [1] p(c_5) = [1] Following rules are strictly oriented: D#(-(x,y)) = [8] x + [8] y + [8] > [8] x + [8] y + [4] = c_3(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(*(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_1(D#(x),D#(y)) D#(+(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_2(D#(x),D#(y)) *** Step 1.b:5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 1.b:5.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,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: D#(*(x,y)) -> c_1(D#(x),D#(y)) The strictly oriented rules are moved into the weak component. **** Step 1.b:5.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,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_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2} Following symbols are considered usable: {D#} TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [2] p(+) = [1] x1 + [1] x2 + [0] p(-) = [1] x1 + [1] x2 + [0] p(0) = [1] p(1) = [1] p(D) = [1] p(constant) = [2] p(t) = [1] p(D#) = [8] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [4] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [0] p(c_5) = [2] Following rules are strictly oriented: D#(*(x,y)) = [8] x + [8] y + [16] > [8] x + [8] y + [4] = c_1(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(+(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_2(D#(x),D#(y)) D#(-(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_3(D#(x),D#(y)) **** Step 1.b:5.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: D#(+(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 1.b:5.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(+(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,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: D#(+(x,y)) -> c_2(D#(x),D#(y)) The strictly oriented rules are moved into the weak component. ***** Step 1.b:5.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(+(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,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_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2} Following symbols are considered usable: {D#} TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [2] p(+) = [1] x1 + [1] x2 + [1] p(-) = [1] x1 + [1] x2 + [0] p(0) = [1] p(1) = [1] p(D) = [1] x1 + [1] p(constant) = [1] p(t) = [1] p(D#) = [8] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [14] p(c_2) = [1] x1 + [1] x2 + [4] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] p(c_5) = [1] Following rules are strictly oriented: D#(+(x,y)) = [8] x + [8] y + [8] > [8] x + [8] y + [4] = c_2(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(*(x,y)) = [8] x + [8] y + [16] >= [8] x + [8] y + [14] = c_1(D#(x),D#(y)) D#(-(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_3(D#(x),D#(y)) ***** Step 1.b:5.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 1.b:5.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:D#(*(x,y)) -> c_1(D#(x),D#(y)) -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 2:W:D#(+(x,y)) -> c_2(D#(x),D#(y)) -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 3:W:D#(-(x,y)) -> c_3(D#(x),D#(y)) -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: D#(*(x,y)) -> c_1(D#(x),D#(y)) 3: D#(-(x,y)) -> c_3(D#(x),D#(y)) 2: D#(+(x,y)) -> c_2(D#(x),D#(y)) ***** Step 1.b:5.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))