MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            f(c(s(x),y)) -> g(c(x,y))
            f(s(x)) -> f(id_inc(c(x,x)))
            g(c(x,x)) -> f(x)
            g(c(x,s(y))) -> g(c(y,x))
            g(c(s(x),y)) -> g(c(y,x))
            id_inc(0()) -> 0()
            id_inc(0()) -> s(0())
            id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
            id_inc(s(x)) -> s(id_inc(x))
        - Signature:
            {f/1,g/1,id_inc/1} / {0/0,c/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,id_inc} and constructors {0,c,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(c(s(x),y)) -> c_1(g#(c(x,y)))
          f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
          g#(c(x,x)) -> c_3(f#(x))
          g#(c(x,s(y))) -> c_4(g#(c(y,x)))
          g#(c(s(x),y)) -> c_5(g#(c(y,x)))
          id_inc#(0()) -> c_6()
          id_inc#(0()) -> c_7()
          id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
          id_inc#(s(x)) -> c_9(id_inc#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(c(s(x),y)) -> c_1(g#(c(x,y)))
            f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
            g#(c(x,x)) -> c_3(f#(x))
            g#(c(x,s(y))) -> c_4(g#(c(y,x)))
            g#(c(s(x),y)) -> c_5(g#(c(y,x)))
            id_inc#(0()) -> c_6()
            id_inc#(0()) -> c_7()
            id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
            id_inc#(s(x)) -> c_9(id_inc#(x))
        - Weak TRS:
            f(c(s(x),y)) -> g(c(x,y))
            f(s(x)) -> f(id_inc(c(x,x)))
            g(c(x,x)) -> f(x)
            g(c(x,s(y))) -> g(c(y,x))
            g(c(s(x),y)) -> g(c(y,x))
            id_inc(0()) -> 0()
            id_inc(0()) -> s(0())
            id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
            id_inc(s(x)) -> s(id_inc(x))
        - Signature:
            {f/1,g/1,id_inc/1,f#/1,g#/1,id_inc#/1} / {0/0,c/2,s/1,c_1/1,c_2/2,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,id_inc#} and constructors {0,c,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          id_inc(0()) -> 0()
          id_inc(0()) -> s(0())
          id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
          id_inc(s(x)) -> s(id_inc(x))
          f#(c(s(x),y)) -> c_1(g#(c(x,y)))
          f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
          g#(c(x,x)) -> c_3(f#(x))
          g#(c(x,s(y))) -> c_4(g#(c(y,x)))
          g#(c(s(x),y)) -> c_5(g#(c(y,x)))
          id_inc#(0()) -> c_6()
          id_inc#(0()) -> c_7()
          id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
          id_inc#(s(x)) -> c_9(id_inc#(x))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(c(s(x),y)) -> c_1(g#(c(x,y)))
            f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
            g#(c(x,x)) -> c_3(f#(x))
            g#(c(x,s(y))) -> c_4(g#(c(y,x)))
            g#(c(s(x),y)) -> c_5(g#(c(y,x)))
            id_inc#(0()) -> c_6()
            id_inc#(0()) -> c_7()
            id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
            id_inc#(s(x)) -> c_9(id_inc#(x))
        - Weak TRS:
            id_inc(0()) -> 0()
            id_inc(0()) -> s(0())
            id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
            id_inc(s(x)) -> s(id_inc(x))
        - Signature:
            {f/1,g/1,id_inc/1,f#/1,g#/1,id_inc#/1} / {0/0,c/2,s/1,c_1/1,c_2/2,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,id_inc#} and constructors {0,c,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {6,7}
        by application of
          Pre({6,7}) = {8,9}.
        Here rules are labelled as follows:
          1: f#(c(s(x),y)) -> c_1(g#(c(x,y)))
          2: f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
          3: g#(c(x,x)) -> c_3(f#(x))
          4: g#(c(x,s(y))) -> c_4(g#(c(y,x)))
          5: g#(c(s(x),y)) -> c_5(g#(c(y,x)))
          6: id_inc#(0()) -> c_6()
          7: id_inc#(0()) -> c_7()
          8: id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
          9: id_inc#(s(x)) -> c_9(id_inc#(x))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(c(s(x),y)) -> c_1(g#(c(x,y)))
            f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
            g#(c(x,x)) -> c_3(f#(x))
            g#(c(x,s(y))) -> c_4(g#(c(y,x)))
            g#(c(s(x),y)) -> c_5(g#(c(y,x)))
            id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
            id_inc#(s(x)) -> c_9(id_inc#(x))
        - Weak DPs:
            id_inc#(0()) -> c_6()
            id_inc#(0()) -> c_7()
        - Weak TRS:
            id_inc(0()) -> 0()
            id_inc(0()) -> s(0())
            id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
            id_inc(s(x)) -> s(id_inc(x))
        - Signature:
            {f/1,g/1,id_inc/1,f#/1,g#/1,id_inc#/1} / {0/0,c/2,s/1,c_1/1,c_2/2,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,id_inc#} and constructors {0,c,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(c(s(x),y)) -> c_1(g#(c(x,y)))
             -->_1 g#(c(s(x),y)) -> c_5(g#(c(y,x))):5
             -->_1 g#(c(x,s(y))) -> c_4(g#(c(y,x))):4
             -->_1 g#(c(x,x)) -> c_3(f#(x)):3
          
          2:S:f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
             -->_2 id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y)):6
             -->_1 f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x))):2
             -->_1 f#(c(s(x),y)) -> c_1(g#(c(x,y))):1
          
          3:S:g#(c(x,x)) -> c_3(f#(x))
             -->_1 f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x))):2
             -->_1 f#(c(s(x),y)) -> c_1(g#(c(x,y))):1
          
          4:S:g#(c(x,s(y))) -> c_4(g#(c(y,x)))
             -->_1 g#(c(s(x),y)) -> c_5(g#(c(y,x))):5
             -->_1 g#(c(x,s(y))) -> c_4(g#(c(y,x))):4
             -->_1 g#(c(x,x)) -> c_3(f#(x)):3
          
          5:S:g#(c(s(x),y)) -> c_5(g#(c(y,x)))
             -->_1 g#(c(s(x),y)) -> c_5(g#(c(y,x))):5
             -->_1 g#(c(x,s(y))) -> c_4(g#(c(y,x))):4
             -->_1 g#(c(x,x)) -> c_3(f#(x)):3
          
          6:S:id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
             -->_2 id_inc#(s(x)) -> c_9(id_inc#(x)):7
             -->_1 id_inc#(s(x)) -> c_9(id_inc#(x)):7
             -->_2 id_inc#(0()) -> c_7():9
             -->_1 id_inc#(0()) -> c_7():9
             -->_2 id_inc#(0()) -> c_6():8
             -->_1 id_inc#(0()) -> c_6():8
             -->_2 id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y)):6
             -->_1 id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y)):6
          
          7:S:id_inc#(s(x)) -> c_9(id_inc#(x))
             -->_1 id_inc#(0()) -> c_7():9
             -->_1 id_inc#(0()) -> c_6():8
             -->_1 id_inc#(s(x)) -> c_9(id_inc#(x)):7
             -->_1 id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y)):6
          
          8:W:id_inc#(0()) -> c_6()
             
          
          9:W:id_inc#(0()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: id_inc#(0()) -> c_6()
          9: id_inc#(0()) -> c_7()
* Step 5: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(c(s(x),y)) -> c_1(g#(c(x,y)))
          f#(s(x)) -> c_2(f#(id_inc(c(x,x))),id_inc#(c(x,x)))
          g#(c(x,x)) -> c_3(f#(x))
          g#(c(x,s(y))) -> c_4(g#(c(y,x)))
          g#(c(s(x),y)) -> c_5(g#(c(y,x)))
          id_inc#(c(x,y)) -> c_8(id_inc#(x),id_inc#(y))
          id_inc#(s(x)) -> c_9(id_inc#(x))
      - Weak TRS:
          id_inc(0()) -> 0()
          id_inc(0()) -> s(0())
          id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
          id_inc(s(x)) -> s(id_inc(x))
      - Signature:
          {f/1,g/1,id_inc/1,f#/1,g#/1,id_inc#/1} / {0/0,c/2,s/1,c_1/1,c_2/2,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {f#,g#,id_inc#} and constructors {0,c,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE