/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,gr0,p} and constructors {0,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,gr0,p} and constructors {0,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,gr0,p} and constructors {0,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          gr(x,y){x -> s(x),y -> s(y)} =
            gr(s(x),s(y)) ->^+ gr(x,y)
              = C[gr(x,y) = gr(x,y){}]

** Step 1.b:1: DependencyPairs.  MAYBE
    + Considered Problem:
        - Strict TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,gr0,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#(gr0(x),p(x),y),gr0#(x),p#(x))
          cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y),gr0#(x))
          gr#(0(),x) -> c_4()
          gr#(s(x),0()) -> c_5()
          gr#(s(x),s(y)) -> c_6(gr#(x,y))
          gr0#(0()) -> c_7()
          gr0#(s(x)) -> c_8()
          p#(0()) -> c_9()
          p#(s(x)) -> c_10()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: 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#(gr0(x),p(x),y),gr0#(x),p#(x))
            cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y),gr0#(x))
            gr#(0(),x) -> c_4()
            gr#(s(x),0()) -> c_5()
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
            gr0#(0()) -> c_7()
            gr0#(s(x)) -> c_8()
            p#(0()) -> c_9()
            p#(s(x)) -> c_10()
        - Weak TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,5,7,8,9,10}
        by application of
          Pre({4,5,7,8,9,10}) = {1,2,3,6}.
        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#(gr0(x),p(x),y),gr0#(x),p#(x))
          3: cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y),gr0#(x))
          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: gr0#(0()) -> c_7()
          8: gr0#(s(x)) -> c_8()
          9: p#(0()) -> c_9()
          10: p#(s(x)) -> c_10()
** Step 1.b:3: 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#(gr0(x),p(x),y),gr0#(x),p#(x))
            cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y),gr0#(x))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
        - Weak DPs:
            gr#(0(),x) -> c_4()
            gr#(s(x),0()) -> c_5()
            gr0#(0()) -> c_7()
            gr0#(s(x)) -> c_8()
            p#(0()) -> c_9()
            p#(s(x)) -> c_10()
        - Weak TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,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#(gr0(x),y,y),gr0#(x)):3
             -->_1 cond2#(false(),x,y) -> c_2(cond1#(gr0(x),p(x),y),gr0#(x),p#(x)):2
             -->_2 gr#(s(x),0()) -> c_5():6
             -->_2 gr#(0(),x) -> c_4():5
          
          2:S:cond2#(false(),x,y) -> c_2(cond1#(gr0(x),p(x),y),gr0#(x),p#(x))
             -->_3 p#(s(x)) -> c_10():10
             -->_3 p#(0()) -> c_9():9
             -->_2 gr0#(s(x)) -> c_8():8
             -->_2 gr0#(0()) -> c_7():7
             -->_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#(gr0(x),y,y),gr0#(x))
             -->_2 gr0#(s(x)) -> c_8():8
             -->_2 gr0#(0()) -> c_7():7
             -->_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():6
             -->_1 gr#(0(),x) -> c_4():5
             -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4
          
          5:W:gr#(0(),x) -> c_4()
             
          
          6:W:gr#(s(x),0()) -> c_5()
             
          
          7:W:gr0#(0()) -> c_7()
             
          
          8:W:gr0#(s(x)) -> c_8()
             
          
          9:W:p#(0()) -> c_9()
             
          
          10:W:p#(s(x)) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: p#(0()) -> c_9()
          10: p#(s(x)) -> c_10()
          7: gr0#(0()) -> c_7()
          8: gr0#(s(x)) -> c_8()
          5: gr#(0(),x) -> c_4()
          6: gr#(s(x),0()) -> c_5()
** Step 1.b:4: 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#(gr0(x),p(x),y),gr0#(x),p#(x))
            cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y),gr0#(x))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
        - Weak TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,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#(gr0(x),y,y),gr0#(x)):3
             -->_1 cond2#(false(),x,y) -> c_2(cond1#(gr0(x),p(x),y),gr0#(x),p#(x)):2
          
          2:S:cond2#(false(),x,y) -> c_2(cond1#(gr0(x),p(x),y),gr0#(x),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#(gr0(x),y,y),gr0#(x))
             -->_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
          
        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#(gr0(x),p(x),y))
          cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y))
** Step 1.b:5: 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#(gr0(x),p(x),y))
            cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y))
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
        - Weak TRS:
            cond1(true(),x,y) -> cond2(gr(x,y),x,y)
            cond2(false(),x,y) -> cond1(gr0(x),p(x),y)
            cond2(true(),x,y) -> cond1(gr0(x),y,y)
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,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)
          gr0(0()) -> false()
          gr0(s(x)) -> 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#(gr0(x),p(x),y))
          cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y))
          gr#(s(x),s(y)) -> c_6(gr#(x,y))
** Step 1.b:6: DecomposeDG.  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#(gr0(x),p(x),y))
            cond2#(true(),x,y) -> c_3(cond1#(gr0(x),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)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y))
          cond2#(false(),x,y) -> c_2(cond1#(gr0(x),p(x),y))
          cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y))
        and a lower component
          gr#(s(x),s(y)) -> c_6(gr#(x,y))
        Further, following extension rules are added to the lower component.
          cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
          cond1#(true(),x,y) -> gr#(x,y)
          cond2#(false(),x,y) -> cond1#(gr0(x),p(x),y)
          cond2#(true(),x,y) -> cond1#(gr0(x),y,y)
*** Step 1.b:6.a:1: 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#(gr0(x),p(x),y))
            cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y))
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,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))
             -->_1 cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y)):3
             -->_1 cond2#(false(),x,y) -> c_2(cond1#(gr0(x),p(x),y)):2
          
          2:S:cond2#(false(),x,y) -> c_2(cond1#(gr0(x),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#(gr0(x),y,y))
             -->_1 cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y),gr#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y))
*** Step 1.b:6.a:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          cond1#(true(),x,y) -> c_1(cond2#(gr(x,y),x,y))
          cond2#(false(),x,y) -> c_2(cond1#(gr0(x),p(x),y))
          cond2#(true(),x,y) -> c_3(cond1#(gr0(x),y,y))
      - Weak TRS:
          gr(0(),x) -> false()
          gr(s(x),0()) -> true()
          gr(s(x),s(y)) -> gr(x,y)
          gr0(0()) -> false()
          gr0(s(x)) -> true()
          p(0()) -> 0()
          p(s(x)) -> x
      - Signature:
          {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,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_10/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,p#} and constructors {0,false,s
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 1.b:6.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            gr#(s(x),s(y)) -> c_6(gr#(x,y))
        - Weak DPs:
            cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
            cond1#(true(),x,y) -> gr#(x,y)
            cond2#(false(),x,y) -> cond1#(gr0(x),p(x),y)
            cond2#(true(),x,y) -> cond1#(gr0(x),y,y)
        - Weak TRS:
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
            gr(s(x),s(y)) -> gr(x,y)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {gr,cond1#,cond2#,gr#,gr0#,p#}
        TcT has computed the following interpretation:
               p(0) = [0]                  
           p(cond1) = [8] x2 + [1]         
           p(cond2) = [8] x1 + [1] x2 + [0]
           p(false) = [1]                  
              p(gr) = [1]                  
             p(gr0) = [0]                  
               p(p) = [2] x1 + [2]         
               p(s) = [1] x1 + [8]         
            p(true) = [1]                  
          p(cond1#) = [1] x3 + [13]        
          p(cond2#) = [8] x1 + [1] x3 + [5]
             p(gr#) = [1] x2 + [0]         
            p(gr0#) = [1] x1 + [1]         
              p(p#) = [1] x1 + [0]         
             p(c_1) = [4] x2 + [0]         
             p(c_2) = [1] x1 + [0]         
             p(c_3) = [2] x1 + [8]         
             p(c_4) = [1]                  
             p(c_5) = [1]                  
             p(c_6) = [1] x1 + [4]         
             p(c_7) = [1]                  
             p(c_8) = [0]                  
             p(c_9) = [1]                  
            p(c_10) = [1]                  
        
        Following rules are strictly oriented:
        gr#(s(x),s(y)) = [1] y + [8]  
                       > [1] y + [4]  
                       = c_6(gr#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
         cond1#(true(),x,y) =  [1] y + [13]         
                            >= [1] y + [13]         
                            =  cond2#(gr(x,y),x,y)  
        
         cond1#(true(),x,y) =  [1] y + [13]         
                            >= [1] y + [0]          
                            =  gr#(x,y)             
        
        cond2#(false(),x,y) =  [1] y + [13]         
                            >= [1] y + [13]         
                            =  cond1#(gr0(x),p(x),y)
        
         cond2#(true(),x,y) =  [1] y + [13]         
                            >= [1] y + [13]         
                            =  cond1#(gr0(x),y,y)   
        
                  gr(0(),x) =  [1]                  
                            >= [1]                  
                            =  false()              
        
               gr(s(x),0()) =  [1]                  
                            >= [1]                  
                            =  true()               
        
              gr(s(x),s(y)) =  [1]                  
                            >= [1]                  
                            =  gr(x,y)              
        
*** Step 1.b:6.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
            cond1#(true(),x,y) -> gr#(x,y)
            cond2#(false(),x,y) -> cond1#(gr0(x),p(x),y)
            cond2#(true(),x,y) -> cond1#(gr0(x),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)
            gr0(0()) -> false()
            gr0(s(x)) -> true()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {cond1/3,cond2/3,gr/2,gr0/1,p/1,cond1#/3,cond2#/3,gr#/2,gr0#/1,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_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,gr0#,p#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),?)