/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
        - Signature:
            {f/3} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
        - Signature:
            {f/3} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
        - Signature:
            {f/3} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,y,z){z -> s(z)} =
            f(x,y,s(z)) ->^+ s(f(0(),1(),z))
              = C[f(0(),1(),z) = f(x,y,z){x -> 0(),y -> 1()}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
        - Signature:
            {f/3} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          f#(0(),1(),x) -> c_2(f#(s(x),x,x))
** Step 1.b:3: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          
        Consider the set of all dependency pairs
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        BEST_CASE TIME (?,?)
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
*** Step 1.b:3.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and 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_2) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(0) = [0]                  
            p(1) = [1]                  
            p(f) = [1] x2 + [1] x3 + [2]
            p(s) = [1] x1 + [4]         
           p(f#) = [4] x3 + [0]         
          p(c_1) = [1] x1 + [15]        
          p(c_2) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        f#(x,y,s(z)) = [4] z + [16]      
                     > [4] z + [15]      
                     = c_1(f#(0(),1(),z))
        
        
        Following rules are (at-least) weakly oriented:
        f#(0(),1(),x) =  [4] x + [0]      
                      >= [4] x + [0]      
                      =  c_2(f#(s(x),x,x))
        
*** Step 1.b:3.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:3.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
             -->_1 f#(0(),1(),x) -> c_2(f#(s(x),x,x)):2
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
          2:W:f#(0(),1(),x) -> c_2(f#(s(x),x,x))
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
*** Step 1.b:3.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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