/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(?,O(n^1))
* Step 1: Sum.  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:
            innermost 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.  MAYBE
    + 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:
            innermost 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: Ara.  MAYBE
    + 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:
            innermost runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "f") :: ["A"(1) x "A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "false") :: [] -(0)-> "A"(0)
          F (TrsFun "if") :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "le") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "main") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "minus") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "perfectp") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "true") :: [] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          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))
          main(x1) -> perfectp(x1)
        2. Weak:
          
        

** 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:
            innermost runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        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(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: 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(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 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_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost 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(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:3: 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(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()
        - Weak 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_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost 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(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(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
             -->_2 f#(0(),y,s(z),u) -> c_2():5
             -->_2 f#(0(),y,0(),u) -> c_1():4
             -->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_2 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(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.
          6: perfectp#(0()) -> c_5()
          4: f#(0(),y,0(),u) -> c_1()
          5: f#(0(),y,s(z),u) -> c_2()
** Step 1.b:4: 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(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 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_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost 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(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(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
             -->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_2 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(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(f#(x,u,z,u))
** Step 1.b:5: UsableRules.  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(f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Weak 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_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost 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#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
          perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
** Step 1.b:6: RemoveHeads.  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(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_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + Applied Processor:
        RemoveHeads
    + 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(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(f#(x,u,z,u))
           -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
           -->_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(f#(x,u,z,u)):2
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(3,perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))))]
** Step 1.b:7: 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(f#(x,u,z,u))
        - 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_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost 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 any 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) = {1}
        
        Following symbols are considered usable:
          {f#,perfectp#}
        TcT has computed the following interpretation:
                  p(0) = [0]                                    
                  p(f) = [8] x1 + [1] x2 + [2] x3 + [1] x4 + [1]
              p(false) = [2]                                    
                 p(if) = [1] x2 + [0]                           
                 p(le) = [2]                                    
              p(minus) = [15]                                   
           p(perfectp) = [4] x1 + [2]                           
                  p(s) = [1] x1 + [2]                           
               p(true) = [0]                                    
                 p(f#) = [13] x1 + [14] x4 + [1]                
          p(perfectp#) = [2] x1 + [4]                           
                p(c_1) = [0]                                    
                p(c_2) = [1]                                    
                p(c_3) = [1] x1 + [14]                          
                p(c_4) = [1] x1 + [15]                          
                p(c_5) = [0]                                    
                p(c_6) = [1] x1 + [1]                           
        
        Following rules are strictly oriented:
         f#(s(x),0(),z,u) = [14] u + [13] x + [27]      
                          > [14] u + [13] x + [15]      
                          = c_3(f#(x,u,minus(z,s(x)),u))
        
        f#(s(x),s(y),z,u) = [14] u + [13] x + [27]      
                          > [14] u + [13] x + [16]      
                          = c_4(f#(x,u,z,u))            
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:8: EmptyProcessor.  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(f#(x,u,z,u))
        - 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_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost 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(?,O(n^1))