/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(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(s(x),y,y) -> f(y,x,s(x))
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,g/2} / {s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(s(x),y,y) -> f(y,x,s(x))
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,g/2} / {s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
          g#(x,y) -> c_2(x)
          g#(x,y) -> c_3(y)
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
            g#(x,y) -> c_2(x)
            g#(x,y) -> c_3(y)
        - Strict TRS:
            f(s(x),y,y) -> f(y,x,s(x))
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
          g#(x,y) -> c_2(x)
          g#(x,y) -> c_3(y)
* Step 4: PredecessorEstimationCP WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
            g#(x,y) -> c_2(x)
            g#(x,y) -> c_3(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
          2: g#(x,y) -> c_2(x)
          3: g#(x,y) -> c_3(y)
          
        The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
            g#(x,y) -> c_2(x)
            g#(x,y) -> c_3(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(f) = [1] x2 + [1]          
            p(g) = [1] x1 + [0]          
            p(s) = [0]                   
           p(f#) = [8] x2 + [8] x3 + [14]
           p(g#) = [9] x1 + [1] x2 + [9] 
          p(c_1) = [8]                   
          p(c_2) = [1] x1 + [0]          
          p(c_3) = [1] x1 + [0]          
        
        Following rules are strictly oriented:
        f#(s(x),y,y) = [16] y + [14]      
                     > [8]                
                     = c_1(f#(y,x,s(x)))  
        
             g#(x,y) = [9] x + [1] y + [9]
                     > [1] x + [0]        
                     = c_2(x)             
        
             g#(x,y) = [9] x + [1] y + [9]
                     > [1] y + [0]        
                     = c_3(y)             
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
            g#(x,y) -> c_2(x)
            g#(x,y) -> c_3(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
            g#(x,y) -> c_2(x)
            g#(x,y) -> c_3(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
             -->_1 f#(s(x),y,y) -> c_1(f#(y,x,s(x))):1
          
          2:W:g#(x,y) -> c_2(x)
             -->_1 g#(x,y) -> c_3(y):3
             -->_1 g#(x,y) -> c_2(x):2
             -->_1 f#(s(x),y,y) -> c_1(f#(y,x,s(x))):1
          
          3:W:g#(x,y) -> c_3(y)
             -->_1 g#(x,y) -> c_3(y):3
             -->_1 g#(x,y) -> c_2(x):2
             -->_1 f#(s(x),y,y) -> c_1(f#(y,x,s(x))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: g#(x,y) -> c_2(x)
          3: g#(x,y) -> c_3(y)
          1: f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
** Step 4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))