WORST_CASE(?,O(n^4)) * Step 1: DependencyPairs WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) c(x) -> mult(x,mult(x,x)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,c,mult} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs add#(0(),y) -> c_1() add#(s(x),y) -> c_2(add#(x,y)) c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(0(),y) -> c_4() mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(0(),y) -> c_1() add#(s(x),y) -> c_2(add#(x,y)) c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(0(),y) -> c_4() mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) c(x) -> mult(x,mult(x,x)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) add#(0(),y) -> c_1() add#(s(x),y) -> c_2(add#(x,y)) c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(0(),y) -> c_4() mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(0(),y) -> c_1() add#(s(x),y) -> c_2(add#(x,y)) c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(0(),y) -> c_4() mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4} by application of Pre({1,4}) = {2,3,5}. Here rules are labelled as follows: 1: add#(0(),y) -> c_1() 2: add#(s(x),y) -> c_2(add#(x,y)) 3: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) 4: mult#(0(),y) -> c_4() 5: mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak DPs: add#(0(),y) -> c_1() mult#(0(),y) -> c_4() - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#(s(x),y) -> c_2(add#(x,y)) -->_1 add#(0(),y) -> c_1():4 -->_1 add#(s(x),y) -> c_2(add#(x,y)):1 2:S:c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):3 -->_1 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):3 -->_2 mult#(0(),y) -> c_4():5 -->_1 mult#(0(),y) -> c_4():5 3:S:mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) -->_2 mult#(0(),y) -> c_4():5 -->_1 add#(0(),y) -> c_1():4 -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):3 -->_1 add#(s(x),y) -> c_2(add#(x,y)):1 4:W:add#(0(),y) -> c_1() 5:W:mult#(0(),y) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: mult#(0(),y) -> c_4() 4: add#(0(),y) -> c_1() * Step 5: Decompose WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,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: add#(s(x),y) -> c_2(add#(x,y)) - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} Problem (S) - Strict DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak DPs: add#(s(x),y) -> c_2(add#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} ** Step 5.a:1: DecomposeDG WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,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 c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) and a lower component add#(s(x),y) -> c_2(add#(x,y)) Further, following extension rules are added to the lower component. c#(x) -> mult#(x,x) c#(x) -> mult#(x,mult(x,x)) mult#(s(x),y) -> add#(mult(x,y),y) mult#(s(x),y) -> mult#(x,y) *** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,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: mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) Consider the set of all dependency pairs 1: mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) 2: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,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_3) = {1,2}, uargs(c_5) = {1,2} Following symbols are considered usable: {add#,c#,mult#} TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x1 + [11] p(c) = [0] p(mult) = [3] x1 + [2] x2 + [0] p(s) = [1] x1 + [9] p(add#) = [4] p(c#) = [9] x1 + [1] p(mult#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [8] x2 + [0] p(c_4) = [0] p(c_5) = [2] x1 + [1] x2 + [0] Following rules are strictly oriented: mult#(s(x),y) = [1] x + [9] > [1] x + [8] = c_5(add#(mult(x,y),y),mult#(x,y)) Following rules are (at-least) weakly oriented: c#(x) = [9] x + [1] >= [9] x + [0] = c_3(mult#(x,mult(x,x)),mult#(x,x)) **** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 -->_1 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 2:W:mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) 2: mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) **** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.a:1.b:1: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) - Weak DPs: c#(x) -> mult#(x,x) c#(x) -> mult#(x,mult(x,x)) mult#(s(x),y) -> add#(mult(x,y),y) mult#(s(x),y) -> mult#(x,y) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(3)): The following argument positions are considered usable: uargs(c_2) = {1} Following symbols are considered usable: {add,mult,add#,c#,mult#} TcT has computed the following interpretation: p(0) = 0 p(add) = x1 + x2 p(c) = 0 p(mult) = x1*x2 p(s) = 1 + x1 p(add#) = 1 + x1 p(c#) = 1 + x1^2 + x1^3 p(mult#) = 1 + x1*x2 p(c_1) = 0 p(c_2) = x1 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 Following rules are strictly oriented: add#(s(x),y) = 2 + x > 1 + x = c_2(add#(x,y)) Following rules are (at-least) weakly oriented: c#(x) = 1 + x^2 + x^3 >= 1 + x^2 = mult#(x,x) c#(x) = 1 + x^2 + x^3 >= 1 + x^3 = mult#(x,mult(x,x)) mult#(s(x),y) = 1 + x*y + y >= 1 + x*y = add#(mult(x,y),y) mult#(s(x),y) = 1 + x*y + y >= 1 + x*y = mult#(x,y) add(0(),y) = y >= y = y add(s(x),y) = 1 + x + y >= 1 + x + y = s(add(x,y)) mult(0(),y) = 0 >= 0 = 0() mult(s(x),y) = x*y + y >= x*y + y = add(mult(x,y),y) *** Step 5.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#(s(x),y) -> c_2(add#(x,y)) c#(x) -> mult#(x,x) c#(x) -> mult#(x,mult(x,x)) mult#(s(x),y) -> add#(mult(x,y),y) mult#(s(x),y) -> mult#(x,y) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak DPs: add#(s(x),y) -> c_2(add#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 -->_1 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 2:S:mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) -->_1 add#(s(x),y) -> c_2(add#(x,y)):3 -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 3:W:add#(s(x),y) -> c_2(add#(x,y)) -->_1 add#(s(x),y) -> c_2(add#(x,y)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: add#(s(x),y) -> c_2(add#(x,y)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 -->_1 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 2:S:mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)) -->_2 mult#(s(x),y) -> c_5(add#(mult(x,y),y),mult#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mult#(s(x),y) -> c_5(mult#(x,y)) ** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,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: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) 2: mult#(s(x),y) -> c_5(mult#(x,y)) The strictly oriented rules are moved into the weak component. *** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,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_3) = {1,2}, uargs(c_5) = {1} Following symbols are considered usable: {add#,c#,mult#} TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x1 + [12] p(c) = [1] p(mult) = [1] x2 + [10] p(s) = [1] x1 + [2] p(add#) = [4] x1 + [1] x2 + [4] p(c#) = [14] x1 + [4] p(mult#) = [1] x1 + [0] p(c_1) = [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [11] x2 + [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: c#(x) = [14] x + [4] > [12] x + [1] = c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) = [1] x + [2] > [1] x + [0] = c_5(mult#(x,y)) Following rules are (at-least) weakly oriented: *** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) mult#(s(x),y) -> c_5(mult#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) -->_2 mult#(s(x),y) -> c_5(mult#(x,y)):2 -->_1 mult#(s(x),y) -> c_5(mult#(x,y)):2 2:W:mult#(s(x),y) -> c_5(mult#(x,y)) -->_1 mult#(s(x),y) -> c_5(mult#(x,y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: c#(x) -> c_3(mult#(x,mult(x,x)),mult#(x,x)) 2: mult#(s(x),y) -> c_5(mult#(x,y)) *** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) mult(0(),y) -> 0() mult(s(x),y) -> add(mult(x,y),y) - Signature: {add/2,c/1,mult/2,add#/2,c#/1,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,c#,mult#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^4))