* Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),y) -> f(x,s(c(y))) - Signature: {f/2} / {c/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {c,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) f#(s(x),y) -> c_2(f#(x,s(c(y)))) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) f#(s(x),y) -> c_2(f#(x,s(c(y)))) - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),y) -> f(x,s(c(y))) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: f#(s(x),y) -> c_2(f#(x,s(c(y)))) The strictly oriented rules are moved into the weak component. ** Step 2.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) f#(s(x),y) -> c_2(f#(x,s(c(y)))) - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),y) -> f(x,s(c(y))) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_2) = {1} Following symbols are considered usable: {f,f#} TcT has computed the following interpretation: p(c) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(f) = [0 0 0] [1 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [1] p(s) = [0 0 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(f#) = [0 0 1] [1 0 0] [1] [1 0 0] x1 + [0 0 1] x2 + [0] [0 1 0] [1 0 1] [0] p(c_1) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 1 0] [0] p(c_2) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: f#(s(x),y) = [0 0 1] [1 0 0] [2] [0 0 0] x + [0 0 1] y + [0] [0 0 1] [1 0 1] [1] > [0 0 1] [1] [0 0 0] x + [0] [0 0 0] [0] = c_2(f#(x,s(c(y)))) Following rules are (at-least) weakly oriented: f#(x,c(y)) = [0 0 1] [1 0 1] [2] [1 0 0] x + [0 0 1] y + [1] [0 1 0] [1 0 2] [2] >= [0 0 1] [1 0 1] [2] [0 0 0] x + [0 0 0] y + [1] [0 1 0] [1 0 2] [2] = c_1(f#(x,s(f(y,y))),f#(y,y)) f(x,c(y)) = [0 0 0] [1 0 1] [2] [0 0 0] x + [0 0 0] y + [0] [0 0 1] [0 0 0] [1] >= [0 0 0] [1] [0 0 0] x + [0] [0 0 1] [1] = f(x,s(f(y,y))) f(s(x),y) = [0 0 0] [1 0 0] [1] [0 0 0] x + [0 0 0] y + [0] [0 0 1] [0 0 0] [2] >= [0 0 0] [1] [0 0 0] x + [0] [0 0 1] [1] = f(x,s(c(y))) ** Step 2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) - Weak DPs: f#(s(x),y) -> c_2(f#(x,s(c(y)))) - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),y) -> f(x,s(c(y))) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) - Weak DPs: f#(s(x),y) -> c_2(f#(x,s(c(y)))) - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),y) -> f(x,s(c(y))) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) -->_2 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2 -->_1 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2 -->_2 f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)):1 2:W:f#(s(x),y) -> c_2(f#(x,s(c(y)))) -->_1 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: f#(s(x),y) -> c_2(f#(x,s(c(y)))) ** Step 2.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),y) -> f(x,s(c(y))) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)) -->_2 f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(x,c(y)) -> c_1(f#(y,y)) ** Step 2.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(y,y)) - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),y) -> f(x,s(c(y))) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f#(x,c(y)) -> c_1(f#(y,y)) ** Step 2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(y,y)) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,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: f#(x,c(y)) -> c_1(f#(y,y)) The strictly oriented rules are moved into the weak component. *** Step 2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,c(y)) -> c_1(f#(y,y)) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,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_1) = {1} Following symbols are considered usable: {f#} TcT has computed the following interpretation: p(c) = [1] x1 + [1] p(f) = [1] p(s) = [0] p(f#) = [2] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [2] Following rules are strictly oriented: f#(x,c(y)) = [2] y + [2] > [2] y + [0] = c_1(f#(y,y)) Following rules are (at-least) weakly oriented: *** Step 2.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(x,c(y)) -> c_1(f#(y,y)) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(x,c(y)) -> c_1(f#(y,y)) - Signature: {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f#(x,c(y)) -> c_1(f#(y,y)) -->_1 f#(x,c(y)) -> c_1(f#(y,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: f#(x,c(y)) -> c_1(f#(y,y)) *** Step 2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))