MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            and(x,false()) -> false()
            and(x,true()) -> x
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            f(true(),x,y) -> f(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
            plus(n,0()) -> n
            plus(n,s(m)) -> s(plus(n,m))
        - Signature:
            {and/2,double/1,f/3,gt/2,plus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and,double,f,gt,plus} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          and#(x,false()) -> c_1()
          and#(x,true()) -> c_2()
          double#(0()) -> c_3()
          double#(s(x)) -> c_4(double#(x))
          f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                               ,and#(gt(x,y),gt(y,s(s(0()))))
                               ,gt#(x,y)
                               ,gt#(y,s(s(0())))
                               ,plus#(s(0()),x)
                               ,double#(y))
          gt#(0(),v) -> c_6()
          gt#(s(u),0()) -> c_7()
          gt#(s(u),s(v)) -> c_8(gt#(u,v))
          plus#(n,0()) -> c_9()
          plus#(n,s(m)) -> c_10(plus#(n,m))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            and#(x,false()) -> c_1()
            and#(x,true()) -> c_2()
            double#(0()) -> c_3()
            double#(s(x)) -> c_4(double#(x))
            f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                 ,and#(gt(x,y),gt(y,s(s(0()))))
                                 ,gt#(x,y)
                                 ,gt#(y,s(s(0())))
                                 ,plus#(s(0()),x)
                                 ,double#(y))
            gt#(0(),v) -> c_6()
            gt#(s(u),0()) -> c_7()
            gt#(s(u),s(v)) -> c_8(gt#(u,v))
            plus#(n,0()) -> c_9()
            plus#(n,s(m)) -> c_10(plus#(n,m))
        - Weak TRS:
            and(x,false()) -> false()
            and(x,true()) -> x
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            f(true(),x,y) -> f(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
            plus(n,0()) -> n
            plus(n,s(m)) -> s(plus(n,m))
        - Signature:
            {and/2,double/1,f/3,gt/2,plus/2,and#/2,double#/1,f#/3,gt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/0,c_4/1,c_5/6,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and#,double#,f#,gt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          and(x,false()) -> false()
          and(x,true()) -> x
          double(0()) -> 0()
          double(s(x)) -> s(s(double(x)))
          gt(0(),v) -> false()
          gt(s(u),0()) -> true()
          gt(s(u),s(v)) -> gt(u,v)
          plus(n,0()) -> n
          plus(n,s(m)) -> s(plus(n,m))
          and#(x,false()) -> c_1()
          and#(x,true()) -> c_2()
          double#(0()) -> c_3()
          double#(s(x)) -> c_4(double#(x))
          f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                               ,and#(gt(x,y),gt(y,s(s(0()))))
                               ,gt#(x,y)
                               ,gt#(y,s(s(0())))
                               ,plus#(s(0()),x)
                               ,double#(y))
          gt#(0(),v) -> c_6()
          gt#(s(u),0()) -> c_7()
          gt#(s(u),s(v)) -> c_8(gt#(u,v))
          plus#(n,0()) -> c_9()
          plus#(n,s(m)) -> c_10(plus#(n,m))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            and#(x,false()) -> c_1()
            and#(x,true()) -> c_2()
            double#(0()) -> c_3()
            double#(s(x)) -> c_4(double#(x))
            f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                 ,and#(gt(x,y),gt(y,s(s(0()))))
                                 ,gt#(x,y)
                                 ,gt#(y,s(s(0())))
                                 ,plus#(s(0()),x)
                                 ,double#(y))
            gt#(0(),v) -> c_6()
            gt#(s(u),0()) -> c_7()
            gt#(s(u),s(v)) -> c_8(gt#(u,v))
            plus#(n,0()) -> c_9()
            plus#(n,s(m)) -> c_10(plus#(n,m))
        - Weak TRS:
            and(x,false()) -> false()
            and(x,true()) -> x
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
            plus(n,0()) -> n
            plus(n,s(m)) -> s(plus(n,m))
        - Signature:
            {and/2,double/1,f/3,gt/2,plus/2,and#/2,double#/1,f#/3,gt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/0,c_4/1,c_5/6,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and#,double#,f#,gt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,6,7,9}
        by application of
          Pre({1,2,3,6,7,9}) = {4,5,8,10}.
        Here rules are labelled as follows:
          1: and#(x,false()) -> c_1()
          2: and#(x,true()) -> c_2()
          3: double#(0()) -> c_3()
          4: double#(s(x)) -> c_4(double#(x))
          5: f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                  ,and#(gt(x,y),gt(y,s(s(0()))))
                                  ,gt#(x,y)
                                  ,gt#(y,s(s(0())))
                                  ,plus#(s(0()),x)
                                  ,double#(y))
          6: gt#(0(),v) -> c_6()
          7: gt#(s(u),0()) -> c_7()
          8: gt#(s(u),s(v)) -> c_8(gt#(u,v))
          9: plus#(n,0()) -> c_9()
          10: plus#(n,s(m)) -> c_10(plus#(n,m))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            double#(s(x)) -> c_4(double#(x))
            f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                 ,and#(gt(x,y),gt(y,s(s(0()))))
                                 ,gt#(x,y)
                                 ,gt#(y,s(s(0())))
                                 ,plus#(s(0()),x)
                                 ,double#(y))
            gt#(s(u),s(v)) -> c_8(gt#(u,v))
            plus#(n,s(m)) -> c_10(plus#(n,m))
        - Weak DPs:
            and#(x,false()) -> c_1()
            and#(x,true()) -> c_2()
            double#(0()) -> c_3()
            gt#(0(),v) -> c_6()
            gt#(s(u),0()) -> c_7()
            plus#(n,0()) -> c_9()
        - Weak TRS:
            and(x,false()) -> false()
            and(x,true()) -> x
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
            plus(n,0()) -> n
            plus(n,s(m)) -> s(plus(n,m))
        - Signature:
            {and/2,double/1,f/3,gt/2,plus/2,and#/2,double#/1,f#/3,gt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/0,c_4/1,c_5/6,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and#,double#,f#,gt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(0()) -> c_3():7
             -->_1 double#(s(x)) -> c_4(double#(x)):1
          
          2:S:f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                   ,and#(gt(x,y),gt(y,s(s(0()))))
                                   ,gt#(x,y)
                                   ,gt#(y,s(s(0())))
                                   ,plus#(s(0()),x)
                                   ,double#(y))
             -->_5 plus#(n,s(m)) -> c_10(plus#(n,m)):4
             -->_4 gt#(s(u),s(v)) -> c_8(gt#(u,v)):3
             -->_3 gt#(s(u),s(v)) -> c_8(gt#(u,v)):3
             -->_5 plus#(n,0()) -> c_9():10
             -->_3 gt#(s(u),0()) -> c_7():9
             -->_4 gt#(0(),v) -> c_6():8
             -->_3 gt#(0(),v) -> c_6():8
             -->_6 double#(0()) -> c_3():7
             -->_2 and#(x,true()) -> c_2():6
             -->_2 and#(x,false()) -> c_1():5
             -->_1 f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                        ,and#(gt(x,y),gt(y,s(s(0()))))
                                        ,gt#(x,y)
                                        ,gt#(y,s(s(0())))
                                        ,plus#(s(0()),x)
                                        ,double#(y)):2
             -->_6 double#(s(x)) -> c_4(double#(x)):1
          
          3:S:gt#(s(u),s(v)) -> c_8(gt#(u,v))
             -->_1 gt#(s(u),0()) -> c_7():9
             -->_1 gt#(0(),v) -> c_6():8
             -->_1 gt#(s(u),s(v)) -> c_8(gt#(u,v)):3
          
          4:S:plus#(n,s(m)) -> c_10(plus#(n,m))
             -->_1 plus#(n,0()) -> c_9():10
             -->_1 plus#(n,s(m)) -> c_10(plus#(n,m)):4
          
          5:W:and#(x,false()) -> c_1()
             
          
          6:W:and#(x,true()) -> c_2()
             
          
          7:W:double#(0()) -> c_3()
             
          
          8:W:gt#(0(),v) -> c_6()
             
          
          9:W:gt#(s(u),0()) -> c_7()
             
          
          10:W:plus#(n,0()) -> c_9()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: and#(x,false()) -> c_1()
          6: and#(x,true()) -> c_2()
          8: gt#(0(),v) -> c_6()
          9: gt#(s(u),0()) -> c_7()
          10: plus#(n,0()) -> c_9()
          7: double#(0()) -> c_3()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            double#(s(x)) -> c_4(double#(x))
            f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                 ,and#(gt(x,y),gt(y,s(s(0()))))
                                 ,gt#(x,y)
                                 ,gt#(y,s(s(0())))
                                 ,plus#(s(0()),x)
                                 ,double#(y))
            gt#(s(u),s(v)) -> c_8(gt#(u,v))
            plus#(n,s(m)) -> c_10(plus#(n,m))
        - Weak TRS:
            and(x,false()) -> false()
            and(x,true()) -> x
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
            plus(n,0()) -> n
            plus(n,s(m)) -> s(plus(n,m))
        - Signature:
            {and/2,double/1,f/3,gt/2,plus/2,and#/2,double#/1,f#/3,gt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/0,c_4/1,c_5/6,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and#,double#,f#,gt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(s(x)) -> c_4(double#(x)):1
          
          2:S:f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                   ,and#(gt(x,y),gt(y,s(s(0()))))
                                   ,gt#(x,y)
                                   ,gt#(y,s(s(0())))
                                   ,plus#(s(0()),x)
                                   ,double#(y))
             -->_5 plus#(n,s(m)) -> c_10(plus#(n,m)):4
             -->_4 gt#(s(u),s(v)) -> c_8(gt#(u,v)):3
             -->_3 gt#(s(u),s(v)) -> c_8(gt#(u,v)):3
             -->_1 f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                                        ,and#(gt(x,y),gt(y,s(s(0()))))
                                        ,gt#(x,y)
                                        ,gt#(y,s(s(0())))
                                        ,plus#(s(0()),x)
                                        ,double#(y)):2
             -->_6 double#(s(x)) -> c_4(double#(x)):1
          
          3:S:gt#(s(u),s(v)) -> c_8(gt#(u,v))
             -->_1 gt#(s(u),s(v)) -> c_8(gt#(u,v)):3
          
          4:S:plus#(n,s(m)) -> c_10(plus#(n,m))
             -->_1 plus#(n,s(m)) -> c_10(plus#(n,m)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                               ,gt#(x,y)
                               ,gt#(y,s(s(0())))
                               ,plus#(s(0()),x)
                               ,double#(y))
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          double#(s(x)) -> c_4(double#(x))
          f#(true(),x,y) -> c_5(f#(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
                               ,gt#(x,y)
                               ,gt#(y,s(s(0())))
                               ,plus#(s(0()),x)
                               ,double#(y))
          gt#(s(u),s(v)) -> c_8(gt#(u,v))
          plus#(n,s(m)) -> c_10(plus#(n,m))
      - Weak TRS:
          and(x,false()) -> false()
          and(x,true()) -> x
          double(0()) -> 0()
          double(s(x)) -> s(s(double(x)))
          gt(0(),v) -> false()
          gt(s(u),0()) -> true()
          gt(s(u),s(v)) -> gt(u,v)
          plus(n,0()) -> n
          plus(n,s(m)) -> s(plus(n,m))
      - Signature:
          {and/2,double/1,f/3,gt/2,plus/2,and#/2,double#/1,f#/3,gt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
          ,c_3/0,c_4/1,c_5/5,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {and#,double#,f#,gt#,plus#} and constructors {0,false,s
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE