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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> cons(0())
            f(s(0())) -> f(p(s(0())))
            p(s(0())) -> 0()
        - Signature:
            {f/1,p/1} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,p} and constructors {0,cons,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(0()) -> c_1()
          f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
          p#(s(0())) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0()) -> c_1()
            f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
            p#(s(0())) -> c_3()
        - Weak TRS:
            f(0()) -> cons(0())
            f(s(0())) -> f(p(s(0())))
            p(s(0())) -> 0()
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,cons,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          p(s(0())) -> 0()
          f#(0()) -> c_1()
          f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
          p#(s(0())) -> c_3()
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0()) -> c_1()
            f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
            p#(s(0())) -> c_3()
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,cons,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2}.
        Here rules are labelled as follows:
          1: f#(0()) -> c_1()
          2: f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
          3: p#(s(0())) -> c_3()
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
        - Weak DPs:
            f#(0()) -> c_1()
            p#(s(0())) -> c_3()
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
             -->_2 p#(s(0())) -> c_3():3
             -->_1 f#(0()) -> c_1():2
             -->_1 f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))):1
          
          2:W:f#(0()) -> c_1()
             
          
          3:W:p#(s(0())) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(0()) -> c_1()
          3: p#(s(0())) -> c_3()
** Step 1.b:5: SimplifyRHS.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,cons,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
             -->_1 f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(s(0())) -> c_2(f#(p(s(0()))))
** Step 1.b:6: WeightGap.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(0())) -> c_2(f#(p(s(0()))))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,cons,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(f#) = {1},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]         
            p(cons) = [0]         
               p(f) = [0]         
               p(p) = [0]         
               p(s) = [3]         
              p(f#) = [1] x1 + [0]
              p(p#) = [0]         
             p(c_1) = [0]         
             p(c_2) = [1] x1 + [0]
             p(c_3) = [0]         
          
          Following rules are strictly oriented:
          f#(s(0())) = [3]               
                     > [0]               
                     = c_2(f#(p(s(0()))))
          
          
          Following rules are (at-least) weakly oriented:
          p(s(0())) =  [0]
                    >= [0]
                    =  0()
          
        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#(s(0())) -> c_2(f#(p(s(0()))))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,cons,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))