WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y goal(xs,ys) -> mul0(xs,ys) mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2} / {Cons/2,Nil/0,S/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0,goal,mul0} and constructors {Cons,Nil,S} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) add0#(Nil(),y) -> c_2() goal#(xs,ys) -> c_3(mul0#(xs,ys)) mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) mul0#(Nil(),y) -> c_5() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) add0#(Nil(),y) -> c_2() goal#(xs,ys) -> c_3(mul0#(xs,ys)) mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) mul0#(Nil(),y) -> c_5() - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y goal(xs,ys) -> mul0(xs,ys) mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) add0#(Nil(),y) -> c_2() goal#(xs,ys) -> c_3(mul0#(xs,ys)) mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) mul0#(Nil(),y) -> c_5() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) add0#(Nil(),y) -> c_2() goal#(xs,ys) -> c_3(mul0#(xs,ys)) mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) mul0#(Nil(),y) -> c_5() - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,5} by application of Pre({2,5}) = {1,3,4}. Here rules are labelled as follows: 1: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) 2: add0#(Nil(),y) -> c_2() 3: goal#(xs,ys) -> c_3(mul0#(xs,ys)) 4: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) 5: mul0#(Nil(),y) -> c_5() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) goal#(xs,ys) -> c_3(mul0#(xs,ys)) mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak DPs: add0#(Nil(),y) -> c_2() mul0#(Nil(),y) -> c_5() - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) -->_1 add0#(Nil(),y) -> c_2():4 -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1 2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys)) -->_1 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3 -->_1 mul0#(Nil(),y) -> c_5():5 3:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) -->_2 mul0#(Nil(),y) -> c_5():5 -->_1 add0#(Nil(),y) -> c_2():4 -->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3 -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1 4:W:add0#(Nil(),y) -> c_2() 5:W:mul0#(Nil(),y) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: mul0#(Nil(),y) -> c_5() 4: add0#(Nil(),y) -> c_2() * Step 5: RemoveHeads WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) goal#(xs,ys) -> c_3(mul0#(xs,ys)) mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1 2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys)) -->_1 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3 3:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) -->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3 -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):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,goal#(xs,ys) -> c_3(mul0#(xs,ys)))] * Step 6: Decompose WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) - Weak DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} Problem (S) - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} ** Step 6.a:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) - Weak DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) and a lower component add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) Further, following extension rules are added to the lower component. mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y) mul0#(Cons(x,xs),y) -> mul0#(xs,y) *** Step 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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_4) = {1,2} Following symbols are considered usable: {add0#,goal#,mul0#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [4] p(Nil) = [2] p(S) = [8] p(add0) = [1] x2 + [9] p(goal) = [4] x1 + [0] p(mul0) = [1] x1 + [14] p(add0#) = [1] p(goal#) = [1] x1 + [0] p(mul0#) = [4] x1 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [8] x1 + [0] p(c_4) = [1] x1 + [1] x2 + [11] p(c_5) = [0] Following rules are strictly oriented: mul0#(Cons(x,xs),y) = [4] xs + [16] > [4] xs + [12] = c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) Following rules are (at-least) weakly oriented: **** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) -->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) **** Step 6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) - Weak DPs: mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y) mul0#(Cons(x,xs),y) -> mul0#(xs,y) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) - Weak DPs: mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y) mul0#(Cons(x,xs),y) -> mul0#(xs,y) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1} Following symbols are considered usable: {add0,mul0,add0#,goal#,mul0#} TcT has computed the following interpretation: p(Cons) = 1 + x2 p(Nil) = 0 p(S) = 1 p(add0) = 4 + x1 + x2 p(goal) = 1 + x1*x2 + x1^2 + x2 + x2^2 p(mul0) = 2*x1 + 2*x1*x2 + 2*x1^2 p(add0#) = 4 + x1 p(goal#) = 2 + x1 + x1^2 + x2 + x2^2 p(mul0#) = 4 + x1 + 5*x1*x2 + 2*x1^2 + 6*x2 p(c_1) = x1 p(c_2) = 0 p(c_3) = 1 p(c_4) = 1 p(c_5) = 0 Following rules are strictly oriented: add0#(Cons(x,xs),y) = 5 + xs > 4 + xs = c_1(add0#(xs,Cons(S(),y))) Following rules are (at-least) weakly oriented: mul0#(Cons(x,xs),y) = 7 + 5*xs + 5*xs*y + 2*xs^2 + 11*y >= 4 + 2*xs + 2*xs*y + 2*xs^2 = add0#(mul0(xs,y),y) mul0#(Cons(x,xs),y) = 7 + 5*xs + 5*xs*y + 2*xs^2 + 11*y >= 4 + xs + 5*xs*y + 2*xs^2 + 6*y = mul0#(xs,y) add0(Cons(x,xs),y) = 5 + xs + y >= 5 + xs + y = add0(xs,Cons(S(),y)) add0(Nil(),y) = 4 + y >= y = y mul0(Cons(x,xs),y) = 4 + 6*xs + 2*xs*y + 2*xs^2 + 2*y >= 4 + 2*xs + 2*xs*y + 2*xs^2 + y = add0(mul0(xs,y),y) mul0(Nil(),y) = 0 >= 0 = Nil() **** Step 6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y) mul0#(Cons(x,xs),y) -> mul0#(xs,y) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y) mul0#(Cons(x,xs),y) -> mul0#(xs,y) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1 2:W:mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y) -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1 3:W:mul0#(Cons(x,xs),y) -> mul0#(xs,y) -->_1 mul0#(Cons(x,xs),y) -> mul0#(xs,y):3 -->_1 mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: mul0#(Cons(x,xs),y) -> mul0#(xs,y) 2: mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y) 1: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) **** Step 6.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak DPs: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):2 -->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1 2:W:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) -->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)) -->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) ** Step 6.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) - Weak TRS: add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y)) add0(Nil(),y) -> y mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y) mul0(Nil(),y) -> Nil() - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) ** Step 6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) The strictly oriented rules are moved into the weak component. *** Step 6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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_4) = {1} Following symbols are considered usable: {add0#,goal#,mul0#} TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [8] p(Nil) = [0] p(S) = [0] p(add0) = [1] x1 + [2] x2 + [0] p(goal) = [1] x1 + [1] x2 + [0] p(mul0) = [1] x1 + [1] x2 + [0] p(add0#) = [0] p(goal#) = [2] p(mul0#) = [2] x1 + [2] x2 + [15] p(c_1) = [2] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [15] p(c_5) = [1] Following rules are strictly oriented: mul0#(Cons(x,xs),y) = [2] x + [2] xs + [2] y + [31] > [2] xs + [2] y + [30] = c_4(mul0#(xs,y)) Following rules are (at-least) weakly oriented: *** Step 6.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) -->_1 mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)) *** Step 6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))