MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            f(x,y,g(z)) -> g(f(x,y,z))
            f(x,g(y),z) -> g(f(x,y,z))
            f(0(),1(),x) -> f(g(x),g(x),x)
            f(g(x),y,z) -> g(f(x,y,z))
        - Signature:
            {f/3} / {0/0,1/0,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {0,1,g}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          f#(x,y,g(z)) -> c_1(f#(x,y,z))
          f#(x,g(y),z) -> c_2(f#(x,y,z))
          f#(0(),1(),x) -> c_3(f#(g(x),g(x),x))
          f#(g(x),y,z) -> c_4(f#(x,y,z))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(x,y,g(z)) -> c_1(f#(x,y,z))
            f#(x,g(y),z) -> c_2(f#(x,y,z))
            f#(0(),1(),x) -> c_3(f#(g(x),g(x),x))
            f#(g(x),y,z) -> c_4(f#(x,y,z))
        - Strict TRS:
            f(x,y,g(z)) -> g(f(x,y,z))
            f(x,g(y),z) -> g(f(x,y,z))
            f(0(),1(),x) -> f(g(x),g(x),x)
            f(g(x),y,z) -> g(f(x,y,z))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(x,y,g(z)) -> c_1(f#(x,y,z))
          f#(x,g(y),z) -> c_2(f#(x,y,z))
          f#(0(),1(),x) -> c_3(f#(g(x),g(x),x))
          f#(g(x),y,z) -> c_4(f#(x,y,z))
* Step 3: PredecessorEstimationCP MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(x,y,g(z)) -> c_1(f#(x,y,z))
            f#(x,g(y),z) -> c_2(f#(x,y,z))
            f#(0(),1(),x) -> c_3(f#(g(x),g(x),x))
            f#(g(x),y,z) -> c_4(f#(x,y,z))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,g}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(x,y,g(z)) -> c_1(f#(x,y,z))
          
        The strictly oriented rules are moved into the weak component.
** Step 3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,g(z)) -> c_1(f#(x,y,z))
            f#(x,g(y),z) -> c_2(f#(x,y,z))
            f#(0(),1(),x) -> c_3(f#(g(x),g(x),x))
            f#(g(x),y,z) -> c_4(f#(x,y,z))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,g}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and 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},
          uargs(c_3) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {f#}
        TcT has computed the following interpretation:
            p(0) = [0]                           
            p(1) = [0]                           
            p(f) = [4] x1 + [1] x2 + [1] x3 + [0]
            p(g) = [1] x1 + [2]                  
           p(f#) = [2] x3 + [0]                  
          p(c_1) = [1] x1 + [3]                  
          p(c_2) = [1] x1 + [0]                  
          p(c_3) = [1] x1 + [0]                  
          p(c_4) = [1] x1 + [0]                  
        
        Following rules are strictly oriented:
        f#(x,y,g(z)) = [2] z + [4]   
                     > [2] z + [3]   
                     = c_1(f#(x,y,z))
        
        
        Following rules are (at-least) weakly oriented:
         f#(x,g(y),z) =  [2] z + [0]         
                      >= [2] z + [0]         
                      =  c_2(f#(x,y,z))      
        
        f#(0(),1(),x) =  [2] x + [0]         
                      >= [2] x + [0]         
                      =  c_3(f#(g(x),g(x),x))
        
         f#(g(x),y,z) =  [2] z + [0]         
                      >= [2] z + [0]         
                      =  c_4(f#(x,y,z))      
        
** Step 3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(x,g(y),z) -> c_2(f#(x,y,z))
            f#(0(),1(),x) -> c_3(f#(g(x),g(x),x))
            f#(g(x),y,z) -> c_4(f#(x,y,z))
        - Weak DPs:
            f#(x,y,g(z)) -> c_1(f#(x,y,z))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,g}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 3.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(x,g(y),z) -> c_2(f#(x,y,z))
          f#(0(),1(),x) -> c_3(f#(g(x),g(x),x))
          f#(g(x),y,z) -> c_4(f#(x,y,z))
      - Weak DPs:
          f#(x,y,g(z)) -> c_1(f#(x,y,z))
      - Signature:
          {f/3,f#/3} / {0/0,1/0,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,g}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE