*** 1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: 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() Weak DP Rules: Weak TRS Rules: Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0} Obligation: Innermost basic terms: {D}/{*,+,-,0,1,constant,t} Applied Processor: DependencyPairs {dpKind_ = WIDP} Proof: 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. *** 1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP 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() Strict TRS Rules: 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() Weak DP Rules: Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: UsableRules Proof: 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() *** 1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP 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() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: Succeeding Proof: () *** 1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP 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() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: 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() *** 1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP 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)) Strict TRS Rules: Weak DP Rules: D#(constant()) -> c_4() D#(t()) -> c_5() Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: RemoveWeakSuffixes Proof: 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() *** 1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP 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)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,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, greedy = NoGreedy}} Proof: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly: 2: D#(+(x,y)) -> c_2(D#(x),D#(y)) The strictly oriented rules are moved into the weak component. *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP 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)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy} Proof: 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 + [2] p(-) = [1] x1 + [1] x2 + [0] p(0) = [2] p(1) = [1] p(D) = [0] p(constant) = [1] p(t) = [1] p(D#) = [8] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [0] p(c_2) = [1] x1 + [1] x2 + [6] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [2] p(c_5) = [0] Following rules are strictly oriented: D#(+(x,y)) = [8] x + [8] y + [16] > [8] x + [8] y + [6] = c_2(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_3(D#(x),D#(y)) *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) Strict TRS Rules: Weak DP Rules: D#(+(x,y)) -> c_2(D#(x),D#(y)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: Assumption Proof: () *** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) Strict TRS Rules: Weak DP Rules: D#(+(x,y)) -> c_2(D#(x),D#(y)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,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, greedy = NoGreedy}} Proof: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly: 2: D#(-(x,y)) -> c_3(D#(x),D#(y)) The strictly oriented rules are moved into the weak component. *** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) Strict TRS Rules: Weak DP Rules: D#(+(x,y)) -> c_2(D#(x),D#(y)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy} Proof: 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 + [4] p(-) = [1] x1 + [1] x2 + [4] p(0) = [2] p(1) = [2] p(D) = [1] p(constant) = [1] p(t) = [1] p(D#) = [1] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [0] p(c_2) = [1] x1 + [1] x2 + [1] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] p(c_5) = [0] Following rules are strictly oriented: D#(-(x,y)) = [1] x + [1] y + [4] > [1] x + [1] y + [0] = c_3(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(*(x,y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_1(D#(x),D#(y)) D#(+(x,y)) = [1] x + [1] y + [4] >= [1] x + [1] y + [1] = c_2(D#(x),D#(y)) *** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) Strict TRS Rules: Weak DP Rules: D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: Assumption Proof: () *** 1.1.1.1.1.1.2.2 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) Strict TRS Rules: Weak DP Rules: D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,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, greedy = NoGreedy}} Proof: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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. *** 1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) Strict TRS Rules: Weak DP Rules: D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy} Proof: 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) = [4] p(1) = [0] p(D) = [2] x1 + [0] p(constant) = [8] p(t) = [1] p(D#) = [8] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [13] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [0] p(c_5) = [0] Following rules are strictly oriented: D#(*(x,y)) = [8] x + [8] y + [16] > [8] x + [8] y + [13] = 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)) *** 1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP 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)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: Assumption Proof: () *** 1.1.1.1.1.1.2.2.2 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP 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)) Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: RemoveWeakSuffixes Proof: 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)) *** 1.1.1.1.1.1.2.2.2.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: 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 basic terms: {D#}/{*,+,-,0,1,constant,t} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).