/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "cons") :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
          F (TrsFun "cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "empty") :: [] -(0)-> "A"(1)
          F (TrsFun "f") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "g") :: ["A"(1) x "A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1) x "A"(0) x "A"(0)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          f(cons(x,k),l) -> g(k,l,cons(x,k))
          f(empty(),l) -> l
          g(a,b,c) -> f(a,cons(b,c))
          main(x1,x2,x3) -> g(x1,x2,x3)
        2. Weak:
          
        

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          f#(empty(),l) -> c_2()
          g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            f#(empty(),l) -> c_2()
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Weak TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {3}.
        Here rules are labelled as follows:
          1: f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          2: f#(empty(),l) -> c_2()
          3: g#(a,b,c) -> c_3(f#(a,cons(b,c)))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Weak DPs:
            f#(empty(),l) -> c_2()
        - Weak TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
             -->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):2
          
          2:S:g#(a,b,c) -> c_3(f#(a,cons(b,c)))
             -->_1 f#(empty(),l) -> c_2():3
             -->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1
          
          3:W:f#(empty(),l) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: f#(empty(),l) -> c_2()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Weak TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          g#(a,b,c) -> c_3(f#(a,cons(b,c)))
** Step 1.b:5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + 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_1) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
           p(cons) = [1] x1 + [1] x2 + [2]
          p(empty) = [1]                  
              p(f) = [1] x1 + [0]         
              p(g) = [2] x1 + [1] x2 + [1]
             p(f#) = [8] x1 + [0]         
             p(g#) = [8] x1 + [0]         
            p(c_1) = [1] x1 + [8]         
            p(c_2) = [1]                  
            p(c_3) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        f#(cons(x,k),l) = [8] k + [8] x + [16]  
                        > [8] k + [8]           
                        = c_1(g#(k,l,cons(x,k)))
        
        
        Following rules are (at-least) weakly oriented:
        g#(a,b,c) =  [8] a + [0]         
                  >= [8] a + [0]         
                  =  c_3(f#(a,cons(b,c)))
        
** Step 1.b:6: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Weak DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + 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(c_1) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
             p(cons) = [1] x2 + [4]                  
            p(empty) = [1]                           
                p(f) = [1] x1 + [1] x2 + [0]         
                p(g) = [1] x1 + [1] x2 + [1] x3 + [0]
               p(f#) = [4] x1 + [5]                  
               p(g#) = [4] x1 + [8]                  
              p(c_1) = [1] x1 + [3]                  
              p(c_2) = [0]                           
              p(c_3) = [1] x1 + [1]                  
          
          Following rules are strictly oriented:
          g#(a,b,c) = [4] a + [8]         
                    > [4] a + [6]         
                    = c_3(f#(a,cons(b,c)))
          
          
          Following rules are (at-least) weakly oriented:
          f#(cons(x,k),l) =  [4] k + [21]          
                          >= [4] k + [11]          
                          =  c_1(g#(k,l,cons(x,k)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))