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


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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
            -(0(),y) -> 0()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {-/2,p/1} / {0/0,greater/2,if/3,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          -#(x,0()) -> c_1(x)
          -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
          -#(0(),y) -> c_3()
          p#(0()) -> c_4()
          p#(s(x)) -> c_5(x)
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5(x)
        - Strict TRS:
            -(x,0()) -> x
            -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
            -(0(),y) -> 0()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          p(s(x)) -> x
          -#(x,0()) -> c_1(x)
          -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
          -#(0(),y) -> c_3()
          p#(0()) -> c_4()
          p#(s(x)) -> c_5(x)
** Step 1.b:3: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5(x)
        - Strict TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + 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(-#) = {2},
            uargs(c_2) = {3}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(-) = [0]                           
                  p(0) = [0]                           
            p(greater) = [1] x1 + [1] x2 + [0]         
                 p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
                  p(p) = [1] x1 + [0]                  
                  p(s) = [1] x1 + [3]                  
                 p(-#) = [1] x2 + [0]                  
                 p(p#) = [0]                           
                p(c_1) = [0]                           
                p(c_2) = [1] x3 + [0]                  
                p(c_3) = [0]                           
                p(c_4) = [0]                           
                p(c_5) = [0]                           
          
          Following rules are strictly oriented:
          p(s(x)) = [1] x + [3]
                  > [1] x + [0]
                  = x          
          
          
          Following rules are (at-least) weakly oriented:
           -#(x,0()) =  [0]                   
                     >= [0]                   
                     =  c_1(x)                
          
          -#(x,s(y)) =  [1] y + [3]           
                     >= [1] y + [3]           
                     =  c_2(x,y,-#(x,p(s(y))))
          
           -#(0(),y) =  [1] y + [0]           
                     >= [0]                   
                     =  c_3()                 
          
             p#(0()) =  [0]                   
                     >= [0]                   
                     =  c_4()                 
          
            p#(s(x)) =  [0]                   
                     >= [0]                   
                     =  c_5(x)                
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: PredecessorEstimation.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4}
        by application of
          Pre({3,4}) = {1,2,5}.
        Here rules are labelled as follows:
          1: -#(x,0()) -> c_1(x)
          2: -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
          3: -#(0(),y) -> c_3()
          4: p#(0()) -> c_4()
          5: p#(s(x)) -> c_5(x)
** Step 1.b:5: RemoveWeakSuffixes.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            p#(s(x)) -> c_5(x)
        - Weak DPs:
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:-#(x,0()) -> c_1(x)
             -->_1 p#(s(x)) -> c_5(x):3
             -->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_1 p#(0()) -> c_4():5
             -->_1 -#(0(),y) -> c_3():4
             -->_1 -#(x,0()) -> c_1(x):1
          
          2:S:-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
             -->_2 p#(s(x)) -> c_5(x):3
             -->_1 p#(s(x)) -> c_5(x):3
             -->_2 p#(0()) -> c_4():5
             -->_1 p#(0()) -> c_4():5
             -->_3 -#(0(),y) -> c_3():4
             -->_2 -#(0(),y) -> c_3():4
             -->_1 -#(0(),y) -> c_3():4
             -->_3 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_2 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_3 -#(x,0()) -> c_1(x):1
             -->_2 -#(x,0()) -> c_1(x):1
             -->_1 -#(x,0()) -> c_1(x):1
          
          3:S:p#(s(x)) -> c_5(x)
             -->_1 p#(0()) -> c_4():5
             -->_1 -#(0(),y) -> c_3():4
             -->_1 p#(s(x)) -> c_5(x):3
             -->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_1 -#(x,0()) -> c_1(x):1
          
          4:W:-#(0(),y) -> c_3()
             
          
          5:W:p#(0()) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: -#(0(),y) -> c_3()
          5: p#(0()) -> c_4()
** Step 1.b:6: PredecessorEstimationCP.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + 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:
          3: p#(s(x)) -> c_5(x)
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:6.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + 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_2) = {3}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(-) = [0]                           
                p(0) = [0]                           
          p(greater) = [1] x1 + [1] x2 + [0]         
               p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
                p(p) = [1] x1 + [0]                  
                p(s) = [1] x1 + [0]                  
               p(-#) = [0]                           
               p(p#) = [1]                           
              p(c_1) = [0]                           
              p(c_2) = [8] x3 + [0]                  
              p(c_3) = [0]                           
              p(c_4) = [0]                           
              p(c_5) = [0]                           
        
        Following rules are strictly oriented:
        p#(s(x)) = [1]   
                 > [0]   
                 = c_5(x)
        
        
        Following rules are (at-least) weakly oriented:
         -#(x,0()) =  [0]                   
                   >= [0]                   
                   =  c_1(x)                
        
        -#(x,s(y)) =  [0]                   
                   >= [0]                   
                   =  c_2(x,y,-#(x,p(s(y))))
        
           p(s(x)) =  [1] x + [0]           
                   >= [1] x + [0]           
                   =  x                     
        
*** Step 1.b:6.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
        - Weak DPs:
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:6.b:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
        - Weak DPs:
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + 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: -#(x,0()) -> c_1(x)
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:1.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
        - Weak DPs:
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + 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_2) = {3}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(-) = [2] x1 + [0] 
                p(0) = [1]          
          p(greater) = [1] x2 + [2] 
               p(if) = [1] x3 + [2] 
                p(p) = [1] x1 + [15]
                p(s) = [1] x1 + [0] 
               p(-#) = [1]          
               p(p#) = [8] x1 + [6] 
              p(c_1) = [0]          
              p(c_2) = [1] x3 + [0] 
              p(c_3) = [1]          
              p(c_4) = [2]          
              p(c_5) = [8] x1 + [1] 
        
        Following rules are strictly oriented:
        -#(x,0()) = [1]   
                  > [0]   
                  = c_1(x)
        
        
        Following rules are (at-least) weakly oriented:
        -#(x,s(y)) =  [1]                   
                   >= [1]                   
                   =  c_2(x,y,-#(x,p(s(y))))
        
          p#(s(x)) =  [8] x + [6]           
                   >= [8] x + [1]           
                   =  c_5(x)                
        
           p(s(x)) =  [1] x + [15]          
                   >= [1] x + [0]           
                   =  x                     
        
**** Step 1.b:6.b:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
        - Weak DPs:
            -#(x,0()) -> c_1(x)
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

**** Step 1.b:6.b:1.b:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
        - Weak DPs:
            -#(x,0()) -> c_1(x)
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:1.b:1.a:1: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
        - Weak DPs:
            -#(x,0()) -> c_1(x)
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_2) = {3}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(-) = [1 1 1]      [2 0 2]      [2]             
                       [2 2 0] x1 + [2 0 2] x2 + [1]             
                       [2 2 1]      [1 0 2]      [1]             
                p(0) = [1]                                       
                       [0]                                       
                       [0]                                       
          p(greater) = [0 0 0]      [0 0 2]      [1]             
                       [0 0 2] x1 + [0 0 0] x2 + [1]             
                       [0 0 0]      [0 0 0]      [0]             
               p(if) = [0 0 1]      [0 0 0]      [0 0 1]      [0]
                       [0 0 2] x1 + [0 0 1] x2 + [0 0 1] x3 + [0]
                       [0 0 0]      [0 0 0]      [0 0 0]      [0]
                p(p) = [2 3 0]      [0]                          
                       [2 0 0] x1 + [3]                          
                       [0 1 0]      [2]                          
                p(s) = [1 1 0]      [0]                          
                       [0 0 1] x1 + [0]                          
                       [0 0 1]      [3]                          
               p(-#) = [0 0 1]      [0 0 2]      [0]             
                       [0 0 3] x1 + [0 1 0] x2 + [2]             
                       [3 2 0]      [2 1 1]      [2]             
               p(p#) = [0 0 2]      [0]                          
                       [1 0 0] x1 + [1]                          
                       [1 3 1]      [0]                          
              p(c_1) = [0 0 0]      [0]                          
                       [0 0 0] x1 + [1]                          
                       [1 0 0]      [0]                          
              p(c_2) = [0 0 0]      [0 0 0]      [1 0 0]      [0]
                       [0 0 0] x1 + [0 0 1] x2 + [0 0 0] x3 + [0]
                       [3 0 0]      [2 0 0]      [0 0 0]      [0]
              p(c_3) = [2]                                       
                       [0]                                       
                       [2]                                       
              p(c_4) = [0]                                       
                       [2]                                       
                       [0]                                       
              p(c_5) = [0 0 0]      [2]                          
                       [1 0 0] x1 + [1]                          
                       [0 0 0]      [2]                          
        
        Following rules are strictly oriented:
        -#(x,s(y)) = [0 0 1]     [0 0 2]     [6]
                     [0 0 3] x + [0 0 1] y + [2]
                     [3 2 0]     [2 2 2]     [5]
                   > [0 0 1]     [0 0 2]     [4]
                     [0 0 0] x + [0 0 1] y + [0]
                     [3 0 0]     [2 0 0]     [0]
                   = c_2(x,y,-#(x,p(s(y))))     
        
        
        Following rules are (at-least) weakly oriented:
        -#(x,0()) =  [0 0 1]     [0]
                     [0 0 3] x + [2]
                     [3 2 0]     [4]
                  >= [0 0 0]     [0]
                     [0 0 0] x + [1]
                     [1 0 0]     [0]
                  =  c_1(x)         
        
         p#(s(x)) =  [0 0 2]     [6]
                     [1 1 0] x + [1]
                     [1 1 4]     [3]
                  >= [0 0 0]     [2]
                     [1 0 0] x + [1]
                     [0 0 0]     [2]
                  =  c_5(x)         
        
          p(s(x)) =  [2 2 3]     [0]
                     [2 2 0] x + [3]
                     [0 0 1]     [2]
                  >= [1 0 0]     [0]
                     [0 1 0] x + [0]
                     [0 0 1]     [0]
                  =  x              
        
***** Step 1.b:6.b:1.b:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.b:1.b:1.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            -#(x,0()) -> c_1(x)
            -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
            p#(s(x)) -> c_5(x)
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:-#(x,0()) -> c_1(x)
             -->_1 p#(s(x)) -> c_5(x):3
             -->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_1 -#(x,0()) -> c_1(x):1
          
          2:W:-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
             -->_2 p#(s(x)) -> c_5(x):3
             -->_1 p#(s(x)) -> c_5(x):3
             -->_3 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_2 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_3 -#(x,0()) -> c_1(x):1
             -->_2 -#(x,0()) -> c_1(x):1
             -->_1 -#(x,0()) -> c_1(x):1
          
          3:W:p#(s(x)) -> c_5(x)
             -->_1 p#(s(x)) -> c_5(x):3
             -->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
             -->_1 -#(x,0()) -> c_1(x):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: -#(x,0()) -> c_1(x)
          3: p#(s(x)) -> c_5(x)
          2: -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
***** Step 1.b:6.b:1.b:1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))