/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(),x2) -> 0()
            f(S(x),x2) -> f(x2,x)
        - Signature:
            {f/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {0,S}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0(),x2) -> 0()
            f(S(x),x2) -> f(x2,x)
        - Signature:
            {f/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {0,S}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(0(),x2) -> c_1()
          f#(S(x),x2) -> c_2(f#(x2,x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),x2) -> c_1()
            f#(S(x),x2) -> c_2(f#(x2,x))
        - Weak TRS:
            f(0(),x2) -> 0()
            f(S(x),x2) -> f(x2,x)
        - Signature:
            {f/2,f#/2} / {0/0,S/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,S}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2}.
        Here rules are labelled as follows:
          1: f#(0(),x2) -> c_1()
          2: f#(S(x),x2) -> c_2(f#(x2,x))
* Step 4: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(S(x),x2) -> c_2(f#(x2,x))
        - Weak DPs:
            f#(0(),x2) -> c_1()
        - Weak TRS:
            f(0(),x2) -> 0()
            f(S(x),x2) -> f(x2,x)
        - Signature:
            {f/2,f#/2} / {0/0,S/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,S}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(S(x),x2) -> c_2(f#(x2,x))
             -->_1 f#(0(),x2) -> c_1():2
             -->_1 f#(S(x),x2) -> c_2(f#(x2,x)):1
          
          2:W:f#(0(),x2) -> c_1()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(0(),x2) -> c_1()
* Step 5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(S(x),x2) -> c_2(f#(x2,x))
        - Weak TRS:
            f(0(),x2) -> 0()
            f(S(x),x2) -> f(x2,x)
        - Signature:
            {f/2,f#/2} / {0/0,S/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,S}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(S(x),x2) -> c_2(f#(x2,x))
* Step 6: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(S(x),x2) -> c_2(f#(x2,x))
        - Signature:
            {f/2,f#/2} / {0/0,S/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,S}
    + 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_2) = {1}
        
        Following symbols are considered usable:
          {f#}
        TcT has computed the following interpretation:
            p(0) = [1]                  
            p(S) = [1] x1 + [12]        
            p(f) = [4] x1 + [1]         
           p(f#) = [1] x1 + [1] x2 + [4]
          p(c_1) = [1]                  
          p(c_2) = [1] x1 + [5]         
        
        Following rules are strictly oriented:
        f#(S(x),x2) = [1] x + [1] x2 + [16]
                    > [1] x + [1] x2 + [9] 
                    = c_2(f#(x2,x))        
        
        
        Following rules are (at-least) weakly oriented:
        
* Step 7: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(S(x),x2) -> c_2(f#(x2,x))
        - Signature:
            {f/2,f#/2} / {0/0,S/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))