WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {0,activate,f,p,s} and constructors {cons,n__0,n__f,n__s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {0,activate,f,p,s} and constructors {cons,n__0,n__f,n__s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__f(x)} = activate(n__f(x)) ->^+ f(activate(x)) = C[activate(x) = activate(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {0,activate,f,p,s} and constructors {cons,n__0,n__f,n__s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) f#(X) -> c_6() f#(0()) -> c_7(0#()) f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) p#(s(X)) -> c_9() s#(X) -> c_10() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) f#(X) -> c_6() f#(0()) -> c_7(0#()) f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) p#(s(X)) -> c_9() s#(X) -> c_10() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,6,7,8,9,10} by application of Pre({1,2,6,7,8,9,10}) = {3,4,5}. Here rules are labelled as follows: 1: 0#() -> c_1() 2: activate#(X) -> c_2() 3: activate#(n__0()) -> c_3(0#()) 4: activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) 5: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 6: f#(X) -> c_6() 7: f#(0()) -> c_7(0#()) 8: f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) 9: p#(s(X)) -> c_9() 10: s#(X) -> c_10() ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__0()) -> c_3(0#()) activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) - Weak DPs: 0#() -> c_1() activate#(X) -> c_2() f#(X) -> c_6() f#(0()) -> c_7(0#()) f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) p#(s(X)) -> c_9() s#(X) -> c_10() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {2,3}. Here rules are labelled as follows: 1: activate#(n__0()) -> c_3(0#()) 2: activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) 3: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 4: 0#() -> c_1() 5: activate#(X) -> c_2() 6: f#(X) -> c_6() 7: f#(0()) -> c_7(0#()) 8: f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) 9: p#(s(X)) -> c_9() 10: s#(X) -> c_10() ** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) - Weak DPs: 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) f#(X) -> c_6() f#(0()) -> c_7(0#()) f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) p#(s(X)) -> c_9() s#(X) -> c_10() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) -->_2 activate#(n__0()) -> c_3(0#()):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):2 -->_1 f#(X) -> c_6():6 -->_2 activate#(X) -> c_2():4 -->_2 activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)):1 2:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__0()) -> c_3(0#()):5 -->_1 s#(X) -> c_10():10 -->_2 activate#(X) -> c_2():4 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)):1 3:W:0#() -> c_1() 4:W:activate#(X) -> c_2() 5:W:activate#(n__0()) -> c_3(0#()) -->_1 0#() -> c_1():3 6:W:f#(X) -> c_6() 7:W:f#(0()) -> c_7(0#()) 8:W:f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) 9:W:p#(s(X)) -> c_9() 10:W:s#(X) -> c_10() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: p#(s(X)) -> c_9() 8: f#(s(0())) -> c_8(f#(p(s(0()))),p#(s(0())),s#(0()),0#()) 7: f#(0()) -> c_7(0#()) 6: f#(X) -> c_6() 4: activate#(X) -> c_2() 10: s#(X) -> c_10() 5: activate#(n__0()) -> c_3(0#()) 3: 0#() -> c_1() ** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)):1 2:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_4(f#(activate(X)),activate#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__f(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) ** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__f(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__s} + 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_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {0#,activate#,f#,p#,s#} TcT has computed the following interpretation: p(0) = [1] p(activate) = [1] x1 + [2] p(cons) = [1] p(f) = [2] x1 + [1] p(n__0) = [1] p(n__f) = [1] x1 + [0] p(n__s) = [1] x1 + [4] p(p) = [1] x1 + [1] p(s) = [1] x1 + [0] p(0#) = [0] p(activate#) = [4] x1 + [0] p(f#) = [1] p(p#) = [2] x1 + [1] p(s#) = [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [2] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [14] p(c_6) = [0] p(c_7) = [2] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [8] x3 + [0] p(c_9) = [4] p(c_10) = [0] Following rules are strictly oriented: activate#(n__s(X)) = [4] X + [16] > [4] X + [14] = c_5(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__f(X)) = [4] X + [0] >= [4] X + [0] = c_4(activate#(X)) ** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_4(activate#(X)) - Weak DPs: activate#(n__s(X)) -> c_5(activate#(X)) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__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}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [0] p(n__0) = [0] p(n__f) = [1] x1 + [3] p(n__s) = [1] x1 + [0] p(p) = [0] p(s) = [0] p(0#) = [0] p(activate#) = [1] x1 + [0] p(f#) = [0] p(p#) = [0] p(s#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] Following rules are strictly oriented: activate#(n__f(X)) = [1] X + [3] > [1] X + [0] = c_4(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__s(X)) = [1] X + [0] >= [1] X + [0] = c_5(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__f(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Signature: {0/0,activate/1,f/1,p/1,s/1,0#/0,activate#/1,f#/1,p#/1,s#/1} / {cons/2,n__0/0,n__f/1,n__s/1,c_1/0,c_2/0 ,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/4,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,f#,p#,s#} and constructors {cons,n__0,n__f ,n__s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))