MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
            cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(0(),s(x)) -> true()
            neq(s(x),0()) -> true()
            neq(s(x),s(y)) -> neq(x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,neq,p} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
          cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
          cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
          gr#(0(),x) -> c_4()
          gr#(s(x),0()) -> c_5()
          gr#(s(x),s(y)) -> c_6(gr#(x,y))
          neq#(0(),0()) -> c_7()
          neq#(0(),s(x)) -> c_8()
          neq#(s(x),0()) -> c_9()
          neq#(s(x),s(y)) -> c_10(neq#(x,y))
          p#(0()) -> c_11()
          p#(s(x)) -> c_12()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
            cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
            cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
            gr#(0(),x) -> c_4()
            gr#(s(x),0()) -> c_5()
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
            neq#(0(),0()) -> c_7()
            neq#(0(),s(x)) -> c_8()
            neq#(s(x),0()) -> c_9()
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
            p#(0()) -> c_11()
            p#(s(x)) -> c_12()
        - Weak TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
            cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(0(),s(x)) -> true()
            neq(s(x),0()) -> true()
            neq(s(x),s(y)) -> neq(x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/3
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          gr(0(),x) -> false()
          gr(s(x),0()) -> true()
          gr(s(x),s(y)) -> gr(x,y)
          neq(0(),0()) -> false()
          neq(s(x),0()) -> true()
          p(0()) -> 0()
          p(s(x)) -> x
          cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
          cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
          cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
          gr#(0(),x) -> c_4()
          gr#(s(x),0()) -> c_5()
          gr#(s(x),s(y)) -> c_6(gr#(x,y))
          neq#(0(),0()) -> c_7()
          neq#(0(),s(x)) -> c_8()
          neq#(s(x),0()) -> c_9()
          neq#(s(x),s(y)) -> c_10(neq#(x,y))
          p#(0()) -> c_11()
          p#(s(x)) -> c_12()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
            cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
            cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
            gr#(0(),x) -> c_4()
            gr#(s(x),0()) -> c_5()
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
            neq#(0(),0()) -> c_7()
            neq#(0(),s(x)) -> c_8()
            neq#(s(x),0()) -> c_9()
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
            p#(0()) -> c_11()
            p#(s(x)) -> c_12()
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(s(x),0()) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/3
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,5,7,8,9,11,12}
        by application of
          Pre({4,5,7,8,9,11,12}) = {1,2,3,6,10}.
        Here rules are labelled as follows:
          1: cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
          2: cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
          3: cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
          4: gr#(0(),x) -> c_4()
          5: gr#(s(x),0()) -> c_5()
          6: gr#(s(x),s(y)) -> c_6(gr#(x,y))
          7: neq#(0(),0()) -> c_7()
          8: neq#(0(),s(x)) -> c_8()
          9: neq#(s(x),0()) -> c_9()
          10: neq#(s(x),s(y)) -> c_10(neq#(x,y))
          11: p#(0()) -> c_11()
          12: p#(s(x)) -> c_12()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
            cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
            cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Weak DPs:
            gr#(0(),x) -> c_4()
            gr#(s(x),0()) -> c_5()
            neq#(0(),0()) -> c_7()
            neq#(0(),s(x)) -> c_8()
            neq#(s(x),0()) -> c_9()
            p#(0()) -> c_11()
            p#(s(x)) -> c_12()
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(s(x),0()) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/3
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
             -->_2 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4
             -->_1 cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0())):3
             -->_1 cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x)):2
             -->_2 gr#(s(x),0()) -> c_5():7
             -->_2 gr#(0(),x) -> c_4():6
          
          2:S:cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
             -->_3 p#(s(x)) -> c_12():12
             -->_3 p#(0()) -> c_11():11
             -->_2 neq#(s(x),0()) -> c_9():10
             -->_2 neq#(0(),0()) -> c_7():8
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):1
          
          3:S:cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
             -->_2 neq#(s(x),0()) -> c_9():10
             -->_2 neq#(0(),0()) -> c_7():8
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):1
          
          4:S:gr#(s(x),s(y)) -> c_6(gr#(x,y))
             -->_1 gr#(s(x),0()) -> c_5():7
             -->_1 gr#(0(),x) -> c_4():6
             -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4
          
          5:S:neq#(s(x),s(y)) -> c_10(neq#(x,y))
             -->_1 neq#(s(x),0()) -> c_9():10
             -->_1 neq#(0(),s(x)) -> c_8():9
             -->_1 neq#(0(),0()) -> c_7():8
             -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):5
          
          6:W:gr#(0(),x) -> c_4()
             
          
          7:W:gr#(s(x),0()) -> c_5()
             
          
          8:W:neq#(0(),0()) -> c_7()
             
          
          9:W:neq#(0(),s(x)) -> c_8()
             
          
          10:W:neq#(s(x),0()) -> c_9()
             
          
          11:W:p#(0()) -> c_11()
             
          
          12:W:p#(s(x)) -> c_12()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: neq#(0(),s(x)) -> c_8()
          11: p#(0()) -> c_11()
          12: p#(s(x)) -> c_12()
          8: neq#(0(),0()) -> c_7()
          10: neq#(s(x),0()) -> c_9()
          6: gr#(0(),x) -> c_4()
          7: gr#(s(x),0()) -> c_5()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
            cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
            cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(s(x),0()) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/3
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
             -->_2 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4
             -->_1 cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0())):3
             -->_1 cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x)):2
          
          2:S:cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y),neq#(x,0()),p#(x))
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):1
          
          3:S:cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y),neq#(x,0()))
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):1
          
          4:S:gr#(s(x),s(y)) -> c_6(gr#(x,y))
             -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4
          
          5:S:neq#(s(x),s(y)) -> c_10(neq#(x,y))
             -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
          cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
* Step 6: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
            cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
            cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(s(x),0()) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
              cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
              cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
              gr#(s(x),s(y)) -> c_6(gr#(x,y))
          - Weak DPs:
              neq#(s(x),s(y)) -> c_10(neq#(x,y))
          - Weak TRS:
              gr(0(),x) -> false()
              gr(s(x),0()) -> true()
              gr(s(x),s(y)) -> gr(x,y)
              neq(0(),0()) -> false()
              neq(s(x),0()) -> true()
              p(0()) -> 0()
              p(s(x)) -> x
          - Signature:
              {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
              ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
              ,true}
        
        Problem (S)
          - Strict DPs:
              neq#(s(x),s(y)) -> c_10(neq#(x,y))
          - Weak DPs:
              cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
              cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
              cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
              gr#(s(x),s(y)) -> c_6(gr#(x,y))
          - Weak TRS:
              gr(0(),x) -> false()
              gr(s(x),0()) -> true()
              gr(s(x),s(y)) -> gr(x,y)
              neq(0(),0()) -> false()
              neq(s(x),0()) -> true()
              p(0()) -> 0()
              p(s(x)) -> x
          - Signature:
              {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
              ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
              ,true}
** Step 6.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
            cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
            cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
        - Weak DPs:
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(s(x),0()) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
             -->_2 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4
             -->_1 cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y)):3
             -->_1 cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y)):2
          
          2:S:cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):1
          
          3:S:cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):1
          
          4:S:gr#(s(x),s(y)) -> c_6(gr#(x,y))
             -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4
          
          5:W:neq#(s(x),s(y)) -> c_10(neq#(x,y))
             -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: neq#(s(x),s(y)) -> c_10(neq#(x,y))
** Step 6.a:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
          cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
          cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
          gr#(s(x),s(y)) -> c_6(gr#(x,y))
      - Weak TRS:
          gr(0(),x) -> false()
          gr(s(x),0()) -> true()
          gr(s(x),s(y)) -> gr(x,y)
          neq(0(),0()) -> false()
          neq(s(x),0()) -> true()
          p(0()) -> 0()
          p(s(x)) -> x
      - Signature:
          {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
          ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Weak DPs:
            cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
            cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
            cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(s(x),0()) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:neq#(s(x),s(y)) -> c_10(neq#(x,y))
             -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):1
          
          2:W:cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
             -->_2 gr#(s(x),s(y)) -> c_6(gr#(x,y)):5
             -->_1 cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y)):4
             -->_1 cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y)):3
          
          3:W:cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):2
          
          4:W:cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):2
          
          5:W:gr#(s(x),s(y)) -> c_6(gr#(x,y))
             -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
          4: cond2#(true(),x,y) -> c_3(cond1#(neq(x,0()),y,y))
          3: cond2#(false(),x,y) -> c_2(cond1#(neq(x,0()),p(x),y))
          5: gr#(s(x),s(y)) -> c_6(gr#(x,y))
** Step 6.b:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            neq(0(),0()) -> false()
            neq(s(x),0()) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          neq#(s(x),s(y)) -> c_10(neq#(x,y))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + 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: neq#(s(x),s(y)) -> c_10(neq#(x,y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + 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_10) = {1}
        
        Following symbols are considered usable:
          {cond1#,cond2#,gr#,neq#,p#}
        TcT has computed the following interpretation:
               p(0) = [1]                           
           p(cond1) = [1] x2 + [1]                  
           p(cond2) = [1] x1 + [1] x2 + [4]         
           p(false) = [4]                           
              p(gr) = [2] x1 + [2]                  
             p(neq) = [1] x2 + [1]                  
               p(p) = [0]                           
               p(s) = [1] x1 + [2]                  
            p(true) = [2]                           
          p(cond1#) = [4] x1 + [1] x2 + [4] x3 + [2]
          p(cond2#) = [2] x1 + [1] x2 + [1] x3 + [1]
             p(gr#) = [1] x1 + [1] x2 + [0]         
            p(neq#) = [8] x2 + [1]                  
              p(p#) = [2] x1 + [0]                  
             p(c_1) = [1] x1 + [0]                  
             p(c_2) = [0]                           
             p(c_3) = [1] x1 + [1]                  
             p(c_4) = [0]                           
             p(c_5) = [0]                           
             p(c_6) = [2] x1 + [4]                  
             p(c_7) = [1]                           
             p(c_8) = [1]                           
             p(c_9) = [8]                           
            p(c_10) = [1] x1 + [8]                  
            p(c_11) = [2]                           
            p(c_12) = [4]                           
        
        Following rules are strictly oriented:
        neq#(s(x),s(y)) = [8] y + [17]   
                        > [8] y + [9]    
                        = c_10(neq#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            neq#(s(x),s(y)) -> c_10(neq#(x,y))
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:neq#(s(x),s(y)) -> c_10(neq#(x,y))
             -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: neq#(s(x),s(y)) -> c_10(neq#(x,y))
*** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {cond1/3,cond2/3,gr/2,neq/2,p/1,cond1#/3,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE