WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: a(b(x)) -> b(a(x)) a(c(x)) -> x - Signature: {a/1} / {b/1,c/1} - Obligation: innermost runtime complexity wrt. defined symbols {a} and constructors {b,c} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a(b(x)) -> b(a(x)) a(c(x)) -> x - Signature: {a/1} / {b/1,c/1} - Obligation: innermost runtime complexity wrt. defined symbols {a} and constructors {b,c} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a(b(x)) -> b(a(x)) a(c(x)) -> x - Signature: {a/1} / {b/1,c/1} - Obligation: innermost runtime complexity wrt. defined symbols {a} and constructors {b,c} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: a(x){x -> b(x)} = a(b(x)) ->^+ b(a(x)) = C[a(x) = a(x){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a(b(x)) -> b(a(x)) a(c(x)) -> x - Signature: {a/1} / {b/1,c/1} - Obligation: innermost runtime complexity wrt. defined symbols {a} and constructors {b,c} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a#(b(x)) -> c_1(a#(x)) a#(c(x)) -> c_2() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a#(b(x)) -> c_1(a#(x)) a#(c(x)) -> c_2() - Weak TRS: a(b(x)) -> b(a(x)) a(c(x)) -> x - Signature: {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {1}. Here rules are labelled as follows: 1: a#(b(x)) -> c_1(a#(x)) 2: a#(c(x)) -> c_2() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a#(b(x)) -> c_1(a#(x)) - Weak DPs: a#(c(x)) -> c_2() - Weak TRS: a(b(x)) -> b(a(x)) a(c(x)) -> x - Signature: {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a#(b(x)) -> c_1(a#(x)) -->_1 a#(c(x)) -> c_2():2 -->_1 a#(b(x)) -> c_1(a#(x)):1 2:W:a#(c(x)) -> c_2() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: a#(c(x)) -> c_2() ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a#(b(x)) -> c_1(a#(x)) - Weak TRS: a(b(x)) -> b(a(x)) a(c(x)) -> x - Signature: {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: a#(b(x)) -> c_1(a#(x)) ** Step 1.b:5: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a#(b(x)) -> c_1(a#(x)) - Signature: {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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: {a#} TcT has computed the following interpretation: p(a) = [2] x1 + [8] p(b) = [1] x1 + [5] p(c) = [1] p(a#) = [4] x1 + [2] p(c_1) = [1] x1 + [0] p(c_2) = [1] Following rules are strictly oriented: a#(b(x)) = [4] x + [22] > [4] x + [2] = c_1(a#(x)) Following rules are (at-least) weakly oriented: ** Step 1.b:6: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: a#(b(x)) -> c_1(a#(x)) - Signature: {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0} - Obligation: innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))