/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(?,O(n^1))
* Step 1: Sum.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> 1()
            f(s(x)) -> f(g(x,x))
            g(0(),1()) -> s(0())
        - Signature:
            {0/0,f/1,g/2} / {1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0,f,g} and constructors {1,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> 1()
            f(s(x)) -> f(g(x,x))
            g(0(),1()) -> s(0())
        - Signature:
            {0/0,f/1,g/2} / {1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0,f,g} and constructors {1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          0#() -> c_1()
          f#(s(x)) -> c_2(f#(g(x,x)))
          g#(0(),1()) -> c_3(0#())
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            0#() -> c_1()
            f#(s(x)) -> c_2(f#(g(x,x)))
            g#(0(),1()) -> c_3(0#())
        - Strict TRS:
            0() -> 1()
            f(s(x)) -> f(g(x,x))
            g(0(),1()) -> s(0())
        - Signature:
            {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0#,f#,g#} and constructors {1,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          0() -> 1()
          g(0(),1()) -> s(0())
          0#() -> c_1()
          f#(s(x)) -> c_2(f#(g(x,x)))
* Step 4: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            0#() -> c_1()
            f#(s(x)) -> c_2(f#(g(x,x)))
        - Strict TRS:
            0() -> 1()
            g(0(),1()) -> s(0())
        - Signature:
            {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0#,f#,g#} and constructors {1,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: 0#() -> c_1()
          2: f#(s(x)) -> c_2(f#(g(x,x)))
* Step 5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_2(f#(g(x,x)))
        - Strict TRS:
            0() -> 1()
            g(0(),1()) -> s(0())
        - Weak DPs:
            0#() -> c_1()
        - Signature:
            {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0#,f#,g#} and constructors {1,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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(0) = [2]         
            p(1) = [0]         
            p(f) = [0]         
            p(g) = [0]         
            p(s) = [0]         
           p(0#) = [0]         
           p(f#) = [0]         
           p(g#) = [0]         
          p(c_1) = [0]         
          p(c_2) = [0]         
          p(c_3) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        0() = [2]
            > [0]
            = 1()
        
        
        Following rules are (at-least) weakly oriented:
              0#() =  [0]            
                   >= [0]            
                   =  c_1()          
        
          f#(s(x)) =  [0]            
                   >= [0]            
                   =  c_2(f#(g(x,x)))
        
        g(0(),1()) =  [0]            
                   >= [0]            
                   =  s(0())         
        
* Step 6: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_2(f#(g(x,x)))
        - Strict TRS:
            g(0(),1()) -> s(0())
        - Weak DPs:
            0#() -> c_1()
        - Weak TRS:
            0() -> 1()
        - Signature:
            {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0#,f#,g#} and constructors {1,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            none
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [0]         
              p(1) = [0]         
              p(f) = [0]         
              p(g) = [0]         
              p(s) = [1] x1 + [0]
             p(0#) = [0]         
             p(f#) = [3]         
             p(g#) = [0]         
            p(c_1) = [0]         
            p(c_2) = [0]         
            p(c_3) = [0]         
          
          Following rules are strictly oriented:
          f#(s(x)) = [3]            
                   > [0]            
                   = c_2(f#(g(x,x)))
          
          
          Following rules are (at-least) weakly oriented:
                0#() =  [0]   
                     >= [0]   
                     =  c_1() 
          
                 0() =  [0]   
                     >= [0]   
                     =  1()   
          
          g(0(),1()) =  [0]   
                     >= [0]   
                     =  s(0())
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            g(0(),1()) -> s(0())
        - Weak DPs:
            0#() -> c_1()
            f#(s(x)) -> c_2(f#(g(x,x)))
        - Weak TRS:
            0() -> 1()
        - Signature:
            {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0#,f#,g#} and constructors {1,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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(0) = [1]         
            p(1) = [1]         
            p(f) = [0]         
            p(g) = [2] x2 + [0]
            p(s) = [0]         
           p(0#) = [0]         
           p(f#) = [0]         
           p(g#) = [0]         
          p(c_1) = [0]         
          p(c_2) = [0]         
          p(c_3) = [0]         
        
        Following rules are strictly oriented:
        g(0(),1()) = [2]   
                   > [0]   
                   = s(0())
        
        
        Following rules are (at-least) weakly oriented:
            0#() =  [0]            
                 >= [0]            
                 =  c_1()          
        
        f#(s(x)) =  [0]            
                 >= [0]            
                 =  c_2(f#(g(x,x)))
        
             0() =  [1]            
                 >= [1]            
                 =  1()            
        
* Step 8: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            0#() -> c_1()
            f#(s(x)) -> c_2(f#(g(x,x)))
        - Weak TRS:
            0() -> 1()
            g(0(),1()) -> s(0())
        - Signature:
            {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {0#,f#,g#} and constructors {1,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))