WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict 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,mult/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,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)) mult#(0(),y) -> c_3() mult#(s(x),y) -> c_4(add#(mult(x,y),y),mult#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(0(),y) -> c_1() add#(s(x),y) -> c_2(add#(x,y)) mult#(0(),y) -> c_3() mult#(s(x),y) -> c_4(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,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3} by application of Pre({1,3}) = {2,4}. Here rules are labelled as follows: 1: add#(0(),y) -> c_1() 2: add#(s(x),y) -> c_2(add#(x,y)) 3: mult#(0(),y) -> c_3() 4: mult#(s(x),y) -> c_4(add#(mult(x,y),y),mult#(x,y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) mult#(s(x),y) -> c_4(add#(mult(x,y),y),mult#(x,y)) - Weak DPs: add#(0(),y) -> c_1() mult#(0(),y) -> c_3() - 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,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,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():3 -->_1 add#(s(x),y) -> c_2(add#(x,y)):1 2:S:mult#(s(x),y) -> c_4(add#(mult(x,y),y),mult#(x,y)) -->_2 mult#(0(),y) -> c_3():4 -->_1 add#(0(),y) -> c_1():3 -->_2 mult#(s(x),y) -> c_4(add#(mult(x,y),y),mult#(x,y)):2 -->_1 add#(s(x),y) -> c_2(add#(x,y)):1 3:W:add#(0(),y) -> c_1() 4:W:mult#(0(),y) -> c_3() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: mult#(0(),y) -> c_3() 3: add#(0(),y) -> c_1() * Step 4: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) mult#(s(x),y) -> c_4(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,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component mult#(s(x),y) -> c_4(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. mult#(s(x),y) -> add#(mult(x,y),y) mult#(s(x),y) -> mult#(x,y) ** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(s(x),y) -> c_4(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,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mult#(s(x),y) -> c_4(add#(mult(x,y),y),mult#(x,y)) -->_2 mult#(s(x),y) -> c_4(add#(mult(x,y),y),mult#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mult#(s(x),y) -> c_4(mult#(x,y)) ** Step 4.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(s(x),y) -> c_4(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,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: mult#(s(x),y) -> c_4(mult#(x,y)) ** Step 4.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(s(x),y) -> c_4(mult#(x,y)) - Signature: {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: 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: all TcT has computed the following interpretation: p(0) = [0] p(add) = [0] p(mult) = [0] p(s) = [1] x1 + [5] p(add#) = [0] p(mult#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] Following rules are strictly oriented: mult#(s(x),y) = [1] x + [5] > [1] x + [0] = c_4(mult#(x,y)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mult#(s(x),y) -> c_4(mult#(x,y)) - Signature: {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 4.b:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) - Weak DPs: 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,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1} Following symbols are considered usable: {add,mult,add#,mult#} TcT has computed the following interpretation: p(0) = 0 p(add) = x1 + 3*x2 p(mult) = 3*x1*x2 p(s) = 1 + x1 p(add#) = 7 + 2*x1 + x2 + x2^2 p(mult#) = 6 + 2*x1 + 7*x1*x2 + 4*x1^2 + 3*x2 + 4*x2^2 p(c_1) = 0 p(c_2) = x1 p(c_3) = 4 p(c_4) = 0 Following rules are strictly oriented: add#(s(x),y) = 9 + 2*x + y + y^2 > 7 + 2*x + y + y^2 = c_2(add#(x,y)) Following rules are (at-least) weakly oriented: mult#(s(x),y) = 12 + 10*x + 7*x*y + 4*x^2 + 10*y + 4*y^2 >= 7 + 6*x*y + y + y^2 = add#(mult(x,y),y) mult#(s(x),y) = 12 + 10*x + 7*x*y + 4*x^2 + 10*y + 4*y^2 >= 6 + 2*x + 7*x*y + 4*x^2 + 3*y + 4*y^2 = mult#(x,y) add(0(),y) = 3*y >= y = y add(s(x),y) = 1 + x + 3*y >= 1 + x + 3*y = s(add(x,y)) mult(0(),y) = 0 >= 0 = 0() mult(s(x),y) = 3*x*y + 3*y >= 3*x*y + 3*y = add(mult(x,y),y) ** Step 4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#(s(x),y) -> c_2(add#(x,y)) 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,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))