/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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

** Step 1.b:1: DependencyPairs.  MAYBE
    + Considered Problem:
        - Strict TRS:
            if(false(),x,y) -> s(minus(p(x),y))
            if(true(),x,y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,y) -> if(le(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x))
          if#(true(),x,y) -> c_2()
          le#(0(),y) -> c_3()
          le#(s(x),0()) -> c_4()
          le#(s(x),s(y)) -> c_5(le#(x,y))
          minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
          p#(0()) -> c_7()
          p#(s(x)) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x))
            if#(true(),x,y) -> c_2()
            le#(0(),y) -> c_3()
            le#(s(x),0()) -> c_4()
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
            p#(0()) -> c_7()
            p#(s(x)) -> c_8()
        - Weak TRS:
            if(false(),x,y) -> s(minus(p(x),y))
            if(true(),x,y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,y) -> if(le(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,p#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,7,8}
        by application of
          Pre({2,3,4,7,8}) = {1,5,6}.
        Here rules are labelled as follows:
          1: if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x))
          2: if#(true(),x,y) -> c_2()
          3: le#(0(),y) -> c_3()
          4: le#(s(x),0()) -> c_4()
          5: le#(s(x),s(y)) -> c_5(le#(x,y))
          6: minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
          7: p#(0()) -> c_7()
          8: p#(s(x)) -> c_8()
** Step 1.b:3: RemoveWeakSuffixes.  MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
        - Weak DPs:
            if#(true(),x,y) -> c_2()
            le#(0(),y) -> c_3()
            le#(s(x),0()) -> c_4()
            p#(0()) -> c_7()
            p#(s(x)) -> c_8()
        - Weak TRS:
            if(false(),x,y) -> s(minus(p(x),y))
            if(true(),x,y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,y) -> if(le(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,p#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x))
             -->_1 minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y)):3
             -->_2 p#(s(x)) -> c_8():8
             -->_2 p#(0()) -> c_7():7
          
          2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
             -->_1 le#(s(x),0()) -> c_4():6
             -->_1 le#(0(),y) -> c_3():5
             -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
          
          3:S:minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
             -->_2 le#(s(x),0()) -> c_4():6
             -->_2 le#(0(),y) -> c_3():5
             -->_1 if#(true(),x,y) -> c_2():4
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
             -->_1 if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x)):1
          
          4:W:if#(true(),x,y) -> c_2()
             
          
          5:W:le#(0(),y) -> c_3()
             
          
          6:W:le#(s(x),0()) -> c_4()
             
          
          7:W:p#(0()) -> c_7()
             
          
          8:W:p#(s(x)) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: p#(0()) -> c_7()
          8: p#(s(x)) -> c_8()
          4: if#(true(),x,y) -> c_2()
          5: le#(0(),y) -> c_3()
          6: le#(s(x),0()) -> c_4()
** Step 1.b:4: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
        - Weak TRS:
            if(false(),x,y) -> s(minus(p(x),y))
            if(true(),x,y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,y) -> if(le(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,p#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x))
             -->_1 minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y)):3
          
          2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
             -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
          
          3:S:minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
             -->_1 if#(false(),x,y) -> c_1(minus#(p(x),y),p#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          if#(false(),x,y) -> c_1(minus#(p(x),y))
** Step 1.b:5: UsableRules.  MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),x,y) -> c_1(minus#(p(x),y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
        - Weak TRS:
            if(false(),x,y) -> s(minus(p(x),y))
            if(true(),x,y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,y) -> if(le(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,p#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          p(0()) -> 0()
          p(s(x)) -> x
          if#(false(),x,y) -> c_1(minus#(p(x),y))
          le#(s(x),s(y)) -> c_5(le#(x,y))
          minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
** Step 1.b:6: DecomposeDG.  MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),x,y) -> c_1(minus#(p(x),y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
        - Weak TRS:
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,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
          if#(false(),x,y) -> c_1(minus#(p(x),y))
          minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
        and a lower component
          le#(s(x),s(y)) -> c_5(le#(x,y))
        Further, following extension rules are added to the lower component.
          if#(false(),x,y) -> minus#(p(x),y)
          minus#(x,y) -> if#(le(x,y),x,y)
          minus#(x,y) -> le#(x,y)
*** Step 1.b:6.a:1: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),x,y) -> c_1(minus#(p(x),y))
            minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
        - Weak TRS:
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,p#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if#(false(),x,y) -> c_1(minus#(p(x),y))
             -->_1 minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y)):2
          
          2:S:minus#(x,y) -> c_6(if#(le(x,y),x,y),le#(x,y))
             -->_1 if#(false(),x,y) -> c_1(minus#(p(x),y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(x,y) -> c_6(if#(le(x,y),x,y))
*** Step 1.b:6.a:2: WeightGap.  MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),x,y) -> c_1(minus#(p(x),y))
            minus#(x,y) -> c_6(if#(le(x,y),x,y))
        - Weak TRS:
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,p#} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(if#) = {1},
            uargs(minus#) = {1},
            uargs(c_1) = {1},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [1]                  
             p(false) = [0]                  
                p(if) = [0]                  
                p(le) = [0]                  
             p(minus) = [0]                  
                 p(p) = [1] x1 + [0]         
                 p(s) = [1] x1 + [0]         
              p(true) = [0]                  
               p(if#) = [1] x1 + [1] x2 + [0]
               p(le#) = [0]                  
            p(minus#) = [1] x1 + [5]         
                p(p#) = [0]                  
               p(c_1) = [1] x1 + [0]         
               p(c_2) = [0]                  
               p(c_3) = [0]                  
               p(c_4) = [0]                  
               p(c_5) = [0]                  
               p(c_6) = [1] x1 + [0]         
               p(c_7) = [0]                  
               p(c_8) = [0]                  
          
          Following rules are strictly oriented:
          minus#(x,y) = [1] x + [5]          
                      > [1] x + [0]          
                      = c_6(if#(le(x,y),x,y))
          
          
          Following rules are (at-least) weakly oriented:
          if#(false(),x,y) =  [1] x + [0]        
                           >= [1] x + [5]        
                           =  c_1(minus#(p(x),y))
          
                 le(0(),y) =  [0]                
                           >= [0]                
                           =  true()             
          
              le(s(x),0()) =  [0]                
                           >= [0]                
                           =  false()            
          
             le(s(x),s(y)) =  [0]                
                           >= [0]                
                           =  le(x,y)            
          
                    p(0()) =  [1]                
                           >= [1]                
                           =  0()                
          
                   p(s(x)) =  [1] x + [0]        
                           >= [1] x + [0]        
                           =  x                  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          if#(false(),x,y) -> c_1(minus#(p(x),y))
      - Weak DPs:
          minus#(x,y) -> c_6(if#(le(x,y),x,y))
      - Weak TRS:
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          p(0()) -> 0()
          p(s(x)) -> x
      - Signature:
          {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
          ,c_6/1,c_7/0,c_8/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {if#,le#,minus#,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:
            le#(s(x),s(y)) -> c_5(le#(x,y))
        - Weak DPs:
            if#(false(),x,y) -> minus#(p(x),y)
            minus#(x,y) -> if#(le(x,y),x,y)
            minus#(x,y) -> le#(x,y)
        - Weak TRS:
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,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_5) = {1}
        
        Following symbols are considered usable:
          {if#,le#,minus#,p#}
        TcT has computed the following interpretation:
               p(0) = [0]         
           p(false) = [0]         
              p(if) = [0]         
              p(le) = [0]         
           p(minus) = [0]         
               p(p) = [0]         
               p(s) = [1] x1 + [2]
            p(true) = [0]         
             p(if#) = [2] x3 + [1]
             p(le#) = [2] x2 + [0]
          p(minus#) = [2] x2 + [1]
              p(p#) = [8]         
             p(c_1) = [1] x1 + [0]
             p(c_2) = [2]         
             p(c_3) = [0]         
             p(c_4) = [1]         
             p(c_5) = [1] x1 + [1]
             p(c_6) = [1] x2 + [1]
             p(c_7) = [1]         
             p(c_8) = [2]         
        
        Following rules are strictly oriented:
        le#(s(x),s(y)) = [2] y + [4]  
                       > [2] y + [1]  
                       = c_5(le#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        if#(false(),x,y) =  [2] y + [1]     
                         >= [2] y + [1]     
                         =  minus#(p(x),y)  
        
             minus#(x,y) =  [2] y + [1]     
                         >= [2] y + [1]     
                         =  if#(le(x,y),x,y)
        
             minus#(x,y) =  [2] y + [1]     
                         >= [2] y + [0]     
                         =  le#(x,y)        
        
*** Step 1.b:6.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if#(false(),x,y) -> minus#(p(x),y)
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,y) -> if#(le(x,y),x,y)
            minus#(x,y) -> le#(x,y)
        - Weak TRS:
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {if/3,le/2,minus/2,p/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if#,le#,minus#,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),?)