/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:
            append(l1,l2) -> ifappend(l1,l2,l1)
            hd(cons(x,l)) -> x
            ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
            ifappend(l1,l2,nil()) -> l2
            is_empty(cons(x,l)) -> false()
            is_empty(nil()) -> true()
            tl(cons(x,l)) -> l
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
            ,false,nil,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Ara.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            append(l1,l2) -> ifappend(l1,l2,l1)
            hd(cons(x,l)) -> x
            ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
            ifappend(l1,l2,nil()) -> l2
            is_empty(cons(x,l)) -> false()
            is_empty(nil()) -> true()
            tl(cons(x,l)) -> l
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
            ,false,nil,true}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 1, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "append") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "false") :: [] -(0)-> "A"(0)
          F (TrsFun "hd") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "ifappend") :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "is_empty") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "nil") :: [] -(0)-> "A"(0)
          F (TrsFun "tl") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "true") :: [] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          hd(cons(x,l)) -> x
          ifappend(l1,l2,nil()) -> l2
          is_empty(cons(x,l)) -> false()
          is_empty(nil()) -> true()
          tl(cons(x,l)) -> l
        2. Weak:
          append(l1,l2) -> ifappend(l1,l2,l1)
          ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            append(l1,l2) -> ifappend(l1,l2,l1)
            ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
        - Weak TRS:
            hd(cons(x,l)) -> x
            ifappend(l1,l2,nil()) -> l2
            is_empty(cons(x,l)) -> false()
            is_empty(nil()) -> true()
            tl(cons(x,l)) -> l
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
            ,false,nil,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(cons) = {2}
        
        Following symbols are considered usable:
          {append,hd,ifappend,is_empty,tl}
        TcT has computed the following interpretation:
            p(append) = [4] x1 + [1] x2 + [4]
              p(cons) = [1] x1 + [1] x2 + [4]
             p(false) = [5]                  
                p(hd) = [4] x1 + [0]         
          p(ifappend) = [1] x2 + [4] x3 + [0]
          p(is_empty) = [1] x1 + [1]         
               p(nil) = [0]                  
                p(tl) = [1] x1 + [1]         
              p(true) = [0]                  
        
        Following rules are strictly oriented:
                    append(l1,l2) = [4] l1 + [1] l2 + [4]        
                                  > [4] l1 + [1] l2 + [0]        
                                  = ifappend(l1,l2,l1)           
        
        ifappend(l1,l2,cons(x,l)) = [4] l + [1] l2 + [4] x + [16]
                                  > [4] l + [1] l2 + [1] x + [8] 
                                  = cons(x,append(l,l2))         
        
        
        Following rules are (at-least) weakly oriented:
                hd(cons(x,l)) =  [4] l + [4] x + [16]
                              >= [1] x + [0]         
                              =  x                   
        
        ifappend(l1,l2,nil()) =  [1] l2 + [0]        
                              >= [1] l2 + [0]        
                              =  l2                  
        
          is_empty(cons(x,l)) =  [1] l + [1] x + [5] 
                              >= [5]                 
                              =  false()             
        
              is_empty(nil()) =  [1]                 
                              >= [0]                 
                              =  true()              
        
                tl(cons(x,l)) =  [1] l + [1] x + [5] 
                              >= [1] l + [0]         
                              =  l                   
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            append(l1,l2) -> ifappend(l1,l2,l1)
            hd(cons(x,l)) -> x
            ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
            ifappend(l1,l2,nil()) -> l2
            is_empty(cons(x,l)) -> false()
            is_empty(nil()) -> true()
            tl(cons(x,l)) -> l
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
            ,false,nil,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))