MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            bin(x,0()) -> s(0())
            bin(0(),s(y)) -> 0()
            bin(s(x),s(y)) -> +(bin(x,s(y)),bin(x,y))
        - Signature:
            {bin/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bin} and constructors {+,0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          bin#(x,0()) -> c_1()
          bin#(0(),s(y)) -> c_2()
          bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            bin#(x,0()) -> c_1()
            bin#(0(),s(y)) -> c_2()
            bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
        - Strict TRS:
            bin(x,0()) -> s(0())
            bin(0(),s(y)) -> 0()
            bin(s(x),s(y)) -> +(bin(x,s(y)),bin(x,y))
        - Signature:
            {bin/2,bin#/2} / {+/2,0/0,s/1,c_1/0,c_2/0,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bin#} and constructors {+,0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          bin#(x,0()) -> c_1()
          bin#(0(),s(y)) -> c_2()
          bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            bin#(x,0()) -> c_1()
            bin#(0(),s(y)) -> c_2()
            bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
        - Signature:
            {bin/2,bin#/2} / {+/2,0/0,s/1,c_1/0,c_2/0,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bin#} and constructors {+,0,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}.
        Here rules are labelled as follows:
          1: bin#(x,0()) -> c_1()
          2: bin#(0(),s(y)) -> c_2()
          3: bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
        - Weak DPs:
            bin#(x,0()) -> c_1()
            bin#(0(),s(y)) -> c_2()
        - Signature:
            {bin/2,bin#/2} / {+/2,0/0,s/1,c_1/0,c_2/0,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bin#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
             -->_2 bin#(0(),s(y)) -> c_2():3
             -->_1 bin#(0(),s(y)) -> c_2():3
             -->_2 bin#(x,0()) -> c_1():2
             -->_2 bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y)):1
             -->_1 bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y)):1
          
          2:W:bin#(x,0()) -> c_1()
             
          
          3:W:bin#(0(),s(y)) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: bin#(x,0()) -> c_1()
          3: bin#(0(),s(y)) -> c_2()
* Step 5: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          bin#(s(x),s(y)) -> c_3(bin#(x,s(y)),bin#(x,y))
      - Signature:
          {bin/2,bin#/2} / {+/2,0/0,s/1,c_1/0,c_2/0,c_3/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {bin#} and constructors {+,0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE