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

** Step 1.b:1: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(cons) = {1,2},
          uargs(copy) = {2,3},
          uargs(f) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
             p(0) = 0                         
          p(cons) = x1 + x2                   
          p(copy) = 4 + 5*x1 + x1*x2 + x2 + x3
             p(f) = 3 + x1                    
             p(n) = 0                         
           p(nil) = 0                         
             p(s) = 2 + x1                    
        
        Following rules are strictly oriented:
                      copy(0(),y,z) = 4 + y + z               
                                    > 3 + z                   
                                    = f(z)                    
        
                     copy(s(x),y,z) = 14 + 5*x + x*y + 3*y + z
                                    > 7 + 5*x + x*y + 2*y + z 
                                    = copy(x,y,cons(f(y),z))  
        
        f(cons(f(cons(nil(),y)),z)) = 6 + y + z               
                                    > 4 + y + z               
                                    = copy(n(),y,z)           
        
                   f(cons(nil(),y)) = 3 + y                   
                                    > y                       
                                    = y                       
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))