WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) - Weak TRS: f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1} / {a/2,c/2,c1/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {F,f} and constructors {a,c,c1,g} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) - Weak TRS: f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1} / {a/2,c/2,c1/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {F,f} and constructors {a,c,c1,g} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) - Weak TRS: f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1} / {a/2,c/2,c1/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {F,f} and constructors {a,c,c1,g} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "F") :: ["A"(1, 1, 1)] -(0)-> "A"(0, 0, 0) F (TrsFun "a") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "c") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "c1") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "f") :: ["A"(1, 1, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "g") :: ["A"(2, 2, 1)] -(1)-> "A"(1, 1, 1) F (TrsFun "g") :: ["A"(2, 1, 0)] -(1)-> "A"(1, 1, 0) F (TrsFun "g") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "main") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) main(x1) -> F(x1) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) - Weak TRS: f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1} / {a/2,c/2,c1/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {F,f} and constructors {a,c,c1,g} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: F(x){x -> g(x)} = F(g(x)) ->^+ c(F(a(g(g(f(x))),g(f(x)))),F(x)) = C[F(x) = F(x){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) - Weak TRS: f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1} / {a/2,c/2,c1/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {F,f} and constructors {a,c,c1,g} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) Weak DPs f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)) and mark the set of starting terms. ** Step 1.b:2: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) - Weak DPs: f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)) - Weak TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1,F#/1,f#/1} / {a/2,c/2,c1/2,g/1,c_1/4,c_2/4,c_3/3} - Obligation: innermost runtime complexity wrt. defined symbols {F#,f#} and constructors {a,c,c1,g} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) -->_3 f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)):3 -->_2 f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)):3 -->_4 F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):2 -->_4 F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):1 2:S:F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) -->_3 f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)):3 -->_2 f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)):3 -->_4 F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):2 -->_4 F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):1 3:W:f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)) -->_3 f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)):3 -->_2 f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: f#(g(z0)) -> c_3(f#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0)) ** Step 1.b:3: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) - Weak TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1,F#/1,f#/1} / {a/2,c/2,c1/2,g/1,c_1/4,c_2/4,c_3/3} - Obligation: innermost runtime complexity wrt. defined symbols {F#,f#} and constructors {a,c,c1,g} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) -->_4 F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):2 -->_4 F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):1 2:S:F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)) -->_4 F#(g(z0)) -> c_2(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):2 -->_4 F#(g(z0)) -> c_1(F#(a(g(g(f(z0))),g(f(z0)))),f#(z0),f#(z0),F#(z0)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: F#(g(z0)) -> c_1(F#(z0)) F#(g(z0)) -> c_2(F#(z0)) ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(g(z0)) -> c_1(F#(z0)) F#(g(z0)) -> c_2(F#(z0)) - Weak TRS: F(g(z0)) -> c(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) F(g(z0)) -> c1(F(a(g(g(f(z0))),g(f(z0)))),F(z0)) f(g(z0)) -> f(a(g(g(f(z0))),g(f(z0)))) - Signature: {F/1,f/1,F#/1,f#/1} / {a/2,c/2,c1/2,g/1,c_1/1,c_2/1,c_3/3} - Obligation: innermost runtime complexity wrt. defined symbols {F#,f#} and constructors {a,c,c1,g} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: F#(g(z0)) -> c_1(F#(z0)) F#(g(z0)) -> c_2(F#(z0)) ** Step 1.b:5: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(g(z0)) -> c_1(F#(z0)) F#(g(z0)) -> c_2(F#(z0)) - Signature: {F/1,f/1,F#/1,f#/1} / {a/2,c/2,c1/2,g/1,c_1/1,c_2/1,c_3/3} - Obligation: innermost runtime complexity wrt. defined symbols {F#,f#} and constructors {a,c,c1,g} + 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}, uargs(c_2) = {1} Following symbols are considered usable: {F#,f#} TcT has computed the following interpretation: p(F) = [4] x1 + [8] p(a) = [0] p(c) = [1] x2 + [0] p(c1) = [1] x2 + [1] p(f) = [1] x1 + [0] p(g) = [1] x1 + [2] p(F#) = [1] x1 + [0] p(f#) = [1] x1 + [2] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [2] p(c_3) = [1] Following rules are strictly oriented: F#(g(z0)) = [1] z0 + [2] > [1] z0 + [1] = c_1(F#(z0)) Following rules are (at-least) weakly oriented: F#(g(z0)) = [1] z0 + [2] >= [1] z0 + [2] = c_2(F#(z0)) ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(g(z0)) -> c_2(F#(z0)) - Weak DPs: F#(g(z0)) -> c_1(F#(z0)) - Signature: {F/1,f/1,F#/1,f#/1} / {a/2,c/2,c1/2,g/1,c_1/1,c_2/1,c_3/3} - Obligation: innermost runtime complexity wrt. defined symbols {F#,f#} and constructors {a,c,c1,g} + 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}, uargs(c_2) = {1} Following symbols are considered usable: {F#,f#} TcT has computed the following interpretation: p(F) = [1] x1 + [1] p(a) = [1] x1 + [2] p(c) = [1] p(c1) = [1] x2 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [4] p(F#) = [4] x1 + [4] p(f#) = [1] x1 + [8] p(c_1) = [1] x1 + [15] p(c_2) = [1] x1 + [8] p(c_3) = [1] x3 + [4] Following rules are strictly oriented: F#(g(z0)) = [4] z0 + [20] > [4] z0 + [12] = c_2(F#(z0)) Following rules are (at-least) weakly oriented: F#(g(z0)) = [4] z0 + [20] >= [4] z0 + [19] = c_1(F#(z0)) ** Step 1.b:7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(g(z0)) -> c_1(F#(z0)) F#(g(z0)) -> c_2(F#(z0)) - Signature: {F/1,f/1,F#/1,f#/1} / {a/2,c/2,c1/2,g/1,c_1/1,c_2/1,c_3/3} - Obligation: innermost runtime complexity wrt. defined symbols {F#,f#} and constructors {a,c,c1,g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))