/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(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,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,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,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,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,y){y -> g(y,z)} =
            f(x,g(y,z)) ->^+ g(f(x,y),z)
              = C[f(x,y) = f(x,y){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,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(g) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {f,norm,rem}
        TcT has computed the following interpretation:
             p(0) = [2]                  
             p(f) = [5] x2 + [6]         
             p(g) = [1] x1 + [4]         
           p(nil) = [0]                  
          p(norm) = [4] x1 + [9]         
           p(rem) = [4] x1 + [1] x2 + [1]
             p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
             f(x,g(y,z)) = [5] y + [26]        
                         > [5] y + [10]        
                         = g(f(x,y),z)         
        
              f(x,nil()) = [6]                 
                         > [4]                 
                         = g(nil(),x)          
        
            norm(g(x,y)) = [4] x + [25]        
                         > [4] x + [9]         
                         = s(norm(x))          
        
             norm(nil()) = [9]                 
                         > [2]                 
                         = 0()                 
        
         rem(g(x,y),0()) = [4] x + [19]        
                         > [1] x + [4]         
                         = g(x,y)              
        
        rem(g(x,y),s(z)) = [4] x + [1] z + [17]
                         > [4] x + [1] z + [1] 
                         = rem(x,z)            
        
            rem(nil(),y) = [1] y + [1]         
                         > [0]                 
                         = nil()               
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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