/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(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1,rev} and constructors {cons,empty}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1,rev} and constructors {cons,empty}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1,rev} 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 "empty") :: [] -(0)-> "A"(0)
          F (TrsFun "main") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "r1") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "rev") :: ["A"(1)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          r1(cons(x,k),a) -> r1(k,cons(x,a))
          r1(empty(),a) -> a
          rev(ls) -> r1(ls,empty())
          main(x1) -> rev(x1)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1,rev} and constructors {cons,empty}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          r1(y,z){y -> cons(x,y)} =
            r1(cons(x,y),z) ->^+ r1(y,cons(x,z))
              = C[r1(y,cons(x,z)) = r1(y,z){z -> cons(x,z)}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1,rev} and constructors {cons,empty}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
          r1#(empty(),a) -> c_2()
          rev#(ls) -> c_3(r1#(ls,empty()))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
            r1#(empty(),a) -> c_2()
            rev#(ls) -> c_3(r1#(ls,empty()))
        - Weak TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1,r1#/2,rev#/1} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1#,rev#} 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}) = {1,3}.
        Here rules are labelled as follows:
          1: r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
          2: r1#(empty(),a) -> c_2()
          3: rev#(ls) -> c_3(r1#(ls,empty()))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
            rev#(ls) -> c_3(r1#(ls,empty()))
        - Weak DPs:
            r1#(empty(),a) -> c_2()
        - Weak TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1,r1#/2,rev#/1} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1#,rev#} and constructors {cons,empty}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
             -->_1 r1#(empty(),a) -> c_2():3
             -->_1 r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a))):1
          
          2:S:rev#(ls) -> c_3(r1#(ls,empty()))
             -->_1 r1#(empty(),a) -> c_2():3
             -->_1 r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a))):1
          
          3:W:r1#(empty(),a) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: r1#(empty(),a) -> c_2()
** Step 1.b:4: RemoveHeads.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
            rev#(ls) -> c_3(r1#(ls,empty()))
        - Weak TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1,r1#/2,rev#/1} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1#,rev#} and constructors {cons,empty}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
           -->_1 r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a))):1
        
        2:S:rev#(ls) -> c_3(r1#(ls,empty()))
           -->_1 r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a))):1
        
        
        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).
        
        [(2,rev#(ls) -> c_3(r1#(ls,empty())))]
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
        - Weak TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1,r1#/2,rev#/1} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1#,rev#} and constructors {cons,empty}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
** Step 1.b:6: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
        - Signature:
            {r1/2,rev/1,r1#/2,rev#/1} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1#,rev#} 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}
        
        Following symbols are considered usable:
          {r1#,rev#}
        TcT has computed the following interpretation:
           p(cons) = [1] x1 + [1] x2 + [2]
          p(empty) = [1]                  
             p(r1) = [1] x2 + [8]         
            p(rev) = [2] x1 + [0]         
            p(r1#) = [8] x1 + [1] x2 + [5]
           p(rev#) = [8] x1 + [1]         
            p(c_1) = [1] x1 + [13]        
            p(c_2) = [0]                  
            p(c_3) = [0]                  
        
        Following rules are strictly oriented:
        r1#(cons(x,k),a) = [1] a + [8] k + [8] x + [21]
                         > [1] a + [8] k + [1] x + [20]
                         = c_1(r1#(k,cons(x,a)))       
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:7: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            r1#(cons(x,k),a) -> c_1(r1#(k,cons(x,a)))
        - Signature:
            {r1/2,rev/1,r1#/2,rev#/1} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {r1#,rev#} and constructors {cons,empty}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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