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


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,0(),u,0()){x -> s(s(x))} =
            f(s(s(x)),0(),u,0()) ->^+ f(x,0(),minus(minus(u,s(s(x))),s(x)),0())
              = C[f(x,0(),minus(minus(u,s(s(x))),s(x)),0()) = f(x,0(),u,0()){u -> minus(minus(u,s(s(x))),s(x))}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          f#(0(),y,0(),u) -> c_1()
          f#(0(),y,s(z),u) -> c_2()
          f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
          perfectp#(0()) -> c_5()
          perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),y,0(),u) -> c_1()
            f#(0(),y,s(z),u) -> c_2()
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
            perfectp#(0()) -> c_5()
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Strict TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/4,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(0(),y,0(),u) -> c_1()
          f#(0(),y,s(z),u) -> c_2()
          f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
          perfectp#(0()) -> c_5()
          perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),y,0(),u) -> c_1()
            f#(0(),y,s(z),u) -> c_2()
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
            perfectp#(0()) -> c_5()
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/4,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,5}
        by application of
          Pre({1,2,5}) = {4,6}.
        Here rules are labelled as follows:
          1: f#(0(),y,0(),u) -> c_1()
          2: f#(0(),y,s(z),u) -> c_2()
          3: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          4: f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
          5: perfectp#(0()) -> c_5()
          6: perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Weak DPs:
            f#(0(),y,0(),u) -> c_1()
            f#(0(),y,s(z),u) -> c_2()
            perfectp#(0()) -> c_5()
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/4,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          2:S:f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
             -->_2 perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))):3
             -->_1 perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))):3
             -->_2 perfectp#(0()) -> c_5():6
             -->_1 perfectp#(0()) -> c_5():6
             -->_4 f#(0(),y,s(z),u) -> c_2():5
             -->_2 f#(0(),y,s(z),u) -> c_2():5
             -->_1 f#(0(),y,s(z),u) -> c_2():5
             -->_4 f#(0(),y,0(),u) -> c_1():4
             -->_2 f#(0(),y,0(),u) -> c_1():4
             -->_1 f#(0(),y,0(),u) -> c_1():4
             -->_4 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_2 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_4 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
             -->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
             -->_1 f#(0(),y,s(z),u) -> c_2():5
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
          
          4:W:f#(0(),y,0(),u) -> c_1()
             
          
          5:W:f#(0(),y,s(z),u) -> c_2()
             
          
          6:W:perfectp#(0()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: f#(0(),y,0(),u) -> c_1()
          6: perfectp#(0()) -> c_5()
          5: f#(0(),y,s(z),u) -> c_2()
** Step 1.b:5: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/4,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          2:S:f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
             -->_2 perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))):3
             -->_1 perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))):3
             -->_4 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_2 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_4 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
             -->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
** Step 1.b:6: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,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:
          3: perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(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:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,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_3) = {1},
          uargs(c_4) = {3},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                  p(0) = [0]                           
                  p(f) = [8] x1 + [1] x2 + [1] x4 + [1]
              p(false) = [0]                           
                 p(if) = [1] x1 + [1] x2 + [1]         
                 p(le) = [1] x1 + [2]                  
              p(minus) = [1] x1 + [0]                  
           p(perfectp) = [4] x1 + [8]                  
                  p(s) = [1] x1 + [0]                  
               p(true) = [1]                           
                 p(f#) = [0]                           
          p(perfectp#) = [14]                          
                p(c_1) = [0]                           
                p(c_2) = [0]                           
                p(c_3) = [1] x1 + [0]                  
                p(c_4) = [8] x3 + [0]                  
                p(c_5) = [1]                           
                p(c_6) = [2] x1 + [0]                  
        
        Following rules are strictly oriented:
        perfectp#(s(x)) = [14]                       
                        > [0]                        
                        = c_6(f#(x,s(0()),s(x),s(x)))
        
        
        Following rules are (at-least) weakly oriented:
         f#(s(x),0(),z,u) =  [0]                         
                          >= [0]                         
                          =  c_3(f#(x,u,minus(z,s(x)),u))
        
        f#(s(x),s(y),z,u) =  [0]                         
                          >= [0]                         
                          =  c_4(x,y,f#(x,u,z,u))        
        
*** Step 1.b:6.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
        - Weak DPs:
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + 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^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
        - Weak DPs:
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,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: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          2: f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
          
        Consider the set of all dependency pairs
          1: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          2: f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
          3: perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        BEST_CASE TIME (?,?)
        SPACE(?,?)on application of the dependency pairs
          {1,2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 1.b:6.b:1.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
        - Weak DPs:
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,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_3) = {1},
          uargs(c_4) = {3},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                  p(0) = [8]                                    
                  p(f) = [1] x1 + [1] x2 + [8] x3 + [1] x4 + [1]
              p(false) = [4]                                    
                 p(if) = [1] x1 + [1] x2 + [1] x3 + [4]         
                 p(le) = [1] x1 + [4]                           
              p(minus) = [1] x1 + [1]                           
           p(perfectp) = [4]                                    
                  p(s) = [1] x1 + [9]                           
               p(true) = [1]                                    
                 p(f#) = [1] x1 + [2] x3 + [0]                  
          p(perfectp#) = [3] x1 + [0]                           
                p(c_1) = [2]                                    
                p(c_2) = [0]                                    
                p(c_3) = [1] x1 + [1]                           
                p(c_4) = [1] x3 + [5]                           
                p(c_5) = [1]                                    
                p(c_6) = [1] x1 + [6]                           
        
        Following rules are strictly oriented:
         f#(s(x),0(),z,u) = [1] x + [2] z + [9]         
                          > [1] x + [2] z + [3]         
                          = c_3(f#(x,u,minus(z,s(x)),u))
        
        f#(s(x),s(y),z,u) = [1] x + [2] z + [9]         
                          > [1] x + [2] z + [5]         
                          = c_4(x,y,f#(x,u,z,u))        
        
        
        Following rules are (at-least) weakly oriented:
        perfectp#(s(x)) =  [3] x + [27]               
                        >= [3] x + [24]               
                        =  c_6(f#(x,s(0()),s(x),s(x)))
        
**** Step 1.b:6.b:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + 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: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u)):2
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          2:W:f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
             -->_2 perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))):3
             -->_1 perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))):3
             -->_3 f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u)):2
             -->_2 f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u)):2
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u)):2
             -->_3 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
             -->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          3:W:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
             -->_1 f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          2: f#(s(x),s(y),z,u) -> c_4(x,y,f#(x,u,z,u))
          3: perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
**** Step 1.b:6.b:1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/3,c_5/0
            ,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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