WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            BIN(z0,0()) -> c()
            BIN(0(),s(z0)) -> c1()
            BIN(s(z0),s(z1)) -> c2(BIN(z0,s(z1)))
            BIN(s(z0),s(z1)) -> c3(BIN(z0,z1))
        - Weak TRS:
            bin(z0,0()) -> s(0())
            bin(0(),s(z0)) -> 0()
            bin(s(z0),s(z1)) -> +(bin(z0,s(z1)),bin(z0,z1))
        - Signature:
            {BIN/2,bin/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN,bin} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            BIN(z0,0()) -> c()
            BIN(0(),s(z0)) -> c1()
            BIN(s(z0),s(z1)) -> c2(BIN(z0,s(z1)))
            BIN(s(z0),s(z1)) -> c3(BIN(z0,z1))
        - Weak TRS:
            bin(z0,0()) -> s(0())
            bin(0(),s(z0)) -> 0()
            bin(s(z0),s(z1)) -> +(bin(z0,s(z1)),bin(z0,z1))
        - Signature:
            {BIN/2,bin/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN,bin} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            BIN(z0,0()) -> c()
            BIN(0(),s(z0)) -> c1()
            BIN(s(z0),s(z1)) -> c2(BIN(z0,s(z1)))
            BIN(s(z0),s(z1)) -> c3(BIN(z0,z1))
        - Weak TRS:
            bin(z0,0()) -> s(0())
            bin(0(),s(z0)) -> 0()
            bin(s(z0),s(z1)) -> +(bin(z0,s(z1)),bin(z0,z1))
        - Signature:
            {BIN/2,bin/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN,bin} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          BIN(x,s(y)){x -> s(x)} =
            BIN(s(x),s(y)) ->^+ c2(BIN(x,s(y)))
              = C[BIN(x,s(y)) = BIN(x,s(y)){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            BIN(z0,0()) -> c()
            BIN(0(),s(z0)) -> c1()
            BIN(s(z0),s(z1)) -> c2(BIN(z0,s(z1)))
            BIN(s(z0),s(z1)) -> c3(BIN(z0,z1))
        - Weak TRS:
            bin(z0,0()) -> s(0())
            bin(0(),s(z0)) -> 0()
            bin(s(z0),s(z1)) -> +(bin(z0,s(z1)),bin(z0,z1))
        - Signature:
            {BIN/2,bin/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN,bin} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          BIN#(z0,0()) -> c_1()
          BIN#(0(),s(z0)) -> c_2()
          BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
          BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
        Weak DPs
          bin#(z0,0()) -> c_5()
          bin#(0(),s(z0)) -> c_6()
          bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1))
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            BIN#(z0,0()) -> c_1()
            BIN#(0(),s(z0)) -> c_2()
            BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
            BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
        - Weak DPs:
            bin#(z0,0()) -> c_5()
            bin#(0(),s(z0)) -> c_6()
            bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1))
        - Weak TRS:
            BIN(z0,0()) -> c()
            BIN(0(),s(z0)) -> c1()
            BIN(s(z0),s(z1)) -> c2(BIN(z0,s(z1)))
            BIN(s(z0),s(z1)) -> c3(BIN(z0,z1))
            bin(z0,0()) -> s(0())
            bin(0(),s(z0)) -> 0()
            bin(s(z0),s(z1)) -> +(bin(z0,s(z1)),bin(z0,z1))
        - Signature:
            {BIN/2,bin/2,BIN#/2,bin#/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN#,bin#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3,4}.
        Here rules are labelled as follows:
          1: BIN#(z0,0()) -> c_1()
          2: BIN#(0(),s(z0)) -> c_2()
          3: BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
          4: BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
          5: bin#(z0,0()) -> c_5()
          6: bin#(0(),s(z0)) -> c_6()
          7: bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
            BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
        - Weak DPs:
            BIN#(z0,0()) -> c_1()
            BIN#(0(),s(z0)) -> c_2()
            bin#(z0,0()) -> c_5()
            bin#(0(),s(z0)) -> c_6()
            bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1))
        - Weak TRS:
            BIN(z0,0()) -> c()
            BIN(0(),s(z0)) -> c1()
            BIN(s(z0),s(z1)) -> c2(BIN(z0,s(z1)))
            BIN(s(z0),s(z1)) -> c3(BIN(z0,z1))
            bin(z0,0()) -> s(0())
            bin(0(),s(z0)) -> 0()
            bin(s(z0),s(z1)) -> +(bin(z0,s(z1)),bin(z0,z1))
        - Signature:
            {BIN/2,bin/2,BIN#/2,bin#/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN#,bin#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
             -->_1 BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1)):2
             -->_1 BIN#(0(),s(z0)) -> c_2():4
             -->_1 BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1))):1
          
          2:S:BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
             -->_1 BIN#(0(),s(z0)) -> c_2():4
             -->_1 BIN#(z0,0()) -> c_1():3
             -->_1 BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1)):2
             -->_1 BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1))):1
          
          3:W:BIN#(z0,0()) -> c_1()
             
          
          4:W:BIN#(0(),s(z0)) -> c_2()
             
          
          5:W:bin#(z0,0()) -> c_5()
             
          
          6:W:bin#(0(),s(z0)) -> c_6()
             
          
          7:W:bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1))
             -->_2 bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1)):7
             -->_1 bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1)):7
             -->_2 bin#(0(),s(z0)) -> c_6():6
             -->_1 bin#(0(),s(z0)) -> c_6():6
             -->_2 bin#(z0,0()) -> c_5():5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: bin#(s(z0),s(z1)) -> c_7(bin#(z0,s(z1)),bin#(z0,z1))
          6: bin#(0(),s(z0)) -> c_6()
          5: bin#(z0,0()) -> c_5()
          3: BIN#(z0,0()) -> c_1()
          4: BIN#(0(),s(z0)) -> c_2()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
            BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
        - Weak TRS:
            BIN(z0,0()) -> c()
            BIN(0(),s(z0)) -> c1()
            BIN(s(z0),s(z1)) -> c2(BIN(z0,s(z1)))
            BIN(s(z0),s(z1)) -> c3(BIN(z0,z1))
            bin(z0,0()) -> s(0())
            bin(0(),s(z0)) -> 0()
            bin(s(z0),s(z1)) -> +(bin(z0,s(z1)),bin(z0,z1))
        - Signature:
            {BIN/2,bin/2,BIN#/2,bin#/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN#,bin#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
          BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
** Step 1.b:5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
            BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
        - Signature:
            {BIN/2,bin/2,BIN#/2,bin#/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN#,bin#} and constructors {+,0,c,c1,c2,c3,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_3) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {BIN#,bin#}
        TcT has computed the following interpretation:
             p(+) = [2]         
             p(0) = [1]         
           p(BIN) = [2] x2 + [0]
           p(bin) = [2] x1 + [2]
             p(c) = [0]         
            p(c1) = [1]         
            p(c2) = [2]         
            p(c3) = [1] x1 + [2]
             p(s) = [1] x1 + [1]
          p(BIN#) = [2] x2 + [0]
          p(bin#) = [2]         
           p(c_1) = [1]         
           p(c_2) = [1]         
           p(c_3) = [1] x1 + [0]
           p(c_4) = [1] x1 + [0]
           p(c_5) = [1]         
           p(c_6) = [0]         
           p(c_7) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        BIN#(s(z0),s(z1)) = [2] z1 + [2]    
                          > [2] z1 + [0]    
                          = c_4(BIN#(z0,z1))
        
        
        Following rules are (at-least) weakly oriented:
        BIN#(s(z0),s(z1)) =  [2] z1 + [2]       
                          >= [2] z1 + [2]       
                          =  c_3(BIN#(z0,s(z1)))
        
** Step 1.b:6: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
        - Weak DPs:
            BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
        - Signature:
            {BIN/2,bin/2,BIN#/2,bin#/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN#,bin#} and constructors {+,0,c,c1,c2,c3,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_3) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(+) = [1] x1 + [1] x2 + [0]
               p(0) = [0]                  
             p(BIN) = [0]                  
             p(bin) = [0]                  
               p(c) = [0]                  
              p(c1) = [0]                  
              p(c2) = [1] x1 + [0]         
              p(c3) = [1] x1 + [1]         
               p(s) = [1] x1 + [1]         
            p(BIN#) = [1] x1 + [0]         
            p(bin#) = [1] x1 + [4] x2 + [1]
             p(c_1) = [4]                  
             p(c_2) = [1]                  
             p(c_3) = [1] x1 + [0]         
             p(c_4) = [1] x1 + [1]         
             p(c_5) = [1]                  
             p(c_6) = [0]                  
             p(c_7) = [1] x1 + [1] x2 + [1]
          
          Following rules are strictly oriented:
          BIN#(s(z0),s(z1)) = [1] z0 + [1]       
                            > [1] z0 + [0]       
                            = c_3(BIN#(z0,s(z1)))
          
          
          Following rules are (at-least) weakly oriented:
          BIN#(s(z0),s(z1)) =  [1] z0 + [1]    
                            >= [1] z0 + [1]    
                            =  c_4(BIN#(z0,z1))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            BIN#(s(z0),s(z1)) -> c_3(BIN#(z0,s(z1)))
            BIN#(s(z0),s(z1)) -> c_4(BIN#(z0,z1))
        - Signature:
            {BIN/2,bin/2,BIN#/2,bin#/2} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {BIN#,bin#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))