/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:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(0(),XS) -> XS
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(0(),XS) -> XS
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(0(),XS) -> XS
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          after(y,x){y -> s(y),x -> cons(u,x)} =
            after(s(y),cons(u,x)) ->^+ after(y,x)
              = C[after(y,x) = after(y,x){}]

** Step 1.b:1: NaturalPI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(0(),XS) -> XS
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(after) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(0) = 1              
          p(activate) = 2 + x1         
             p(after) = 3 + 8*x1 + 8*x2
              p(cons) = x2             
              p(from) = 0              
           p(n__from) = 0              
                 p(s) = 2 + x1         
        
        Following rules are strictly oriented:
                 activate(X) = 2 + X    
                             > X        
                             = X        
        
        activate(n__from(X)) = 2        
                             > 0        
                             = from(X)  
        
               after(0(),XS) = 11 + 8*XS
                             > XS       
                             = XS       
        
        
        Following rules are (at-least) weakly oriented:
        after(s(N),cons(X,XS)) =  19 + 8*N + 8*XS      
                               >= 19 + 8*N + 8*XS      
                               =  after(N,activate(XS))
        
                       from(X) =  0                    
                               >= 0                    
                               =  cons(X,n__from(s(X)))
        
                       from(X) =  0                    
                               >= 0                    
                               =  n__from(X)           
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(0(),XS) -> XS
        - Signature:
            {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,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(after) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(0) = [5]                  
          p(activate) = [1] x1 + [0]         
             p(after) = [2] x1 + [4] x2 + [6]
              p(cons) = [1] x2 + [0]         
              p(from) = [0]                  
           p(n__from) = [0]                  
                 p(s) = [1] x1 + [1]         
        
        Following rules are strictly oriented:
        after(s(N),cons(X,XS)) = [2] N + [4] XS + [8] 
                               > [2] N + [4] XS + [6] 
                               = after(N,activate(XS))
        
        
        Following rules are (at-least) weakly oriented:
                 activate(X) =  [1] X + [0]          
                             >= [1] X + [0]          
                             =  X                    
        
        activate(n__from(X)) =  [0]                  
                             >= [0]                  
                             =  from(X)              
        
               after(0(),XS) =  [4] XS + [16]        
                             >= [1] XS + [0]         
                             =  XS                   
        
                     from(X) =  [0]                  
                             >= [0]                  
                             =  cons(X,n__from(s(X)))
        
                     from(X) =  [0]                  
                             >= [0]                  
                             =  n__from(X)           
        
** Step 1.b:3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(0(),XS) -> XS
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
        - Signature:
            {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,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(after) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(0) = [12]                 
          p(activate) = [1] x1 + [2]         
             p(after) = [2] x1 + [8] x2 + [2]
              p(cons) = [1] x2 + [0]         
              p(from) = [1]                  
           p(n__from) = [0]                  
                 p(s) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
        from(X) = [1]                  
                > [0]                  
                = cons(X,n__from(s(X)))
        
        from(X) = [1]                  
                > [0]                  
                = n__from(X)           
        
        
        Following rules are (at-least) weakly oriented:
                   activate(X) =  [1] X + [2]          
                               >= [1] X + [0]          
                               =  X                    
        
          activate(n__from(X)) =  [2]                  
                               >= [1]                  
                               =  from(X)              
        
                 after(0(),XS) =  [8] XS + [26]        
                               >= [1] XS + [0]         
                               =  XS                   
        
        after(s(N),cons(X,XS)) =  [2] N + [8] XS + [18]
                               >= [2] N + [8] XS + [18]
                               =  after(N,activate(XS))
        
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(0(),XS) -> XS
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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