WORST_CASE(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,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(sieve) = {1}
        
        Following symbols are considered usable:
          {filter,nats,sieve,zprimes}
        TcT has computed the following interpretation:
                p(0) = [4]         
             p(cons) = [0]         
           p(filter) = [8] x1 + [8]
             p(nats) = [0]         
                p(s) = [6]         
            p(sieve) = [1] x1 + [0]
          p(zprimes) = [0]         
        
        Following rules are strictly oriented:
         filter(cons(X),0(),M) = [8]      
                               > [0]      
                               = cons(0())
        
        filter(cons(X),s(N),M) = [8]      
                               > [0]      
                               = cons(X)  
        
        
        Following rules are (at-least) weakly oriented:
                  nats(N) =  [0]                   
                          >= [0]                   
                          =  cons(N)               
        
         sieve(cons(0())) =  [0]                   
                          >= [0]                   
                          =  cons(0())             
        
        sieve(cons(s(N))) =  [0]                   
                          >= [0]                   
                          =  cons(s(N))            
        
                zprimes() =  [0]                   
                          >= [0]                   
                          =  sieve(nats(s(s(0()))))
        
* Step 3: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,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(sieve) = {1}
        
        Following symbols are considered usable:
          {filter,nats,sieve,zprimes}
        TcT has computed the following interpretation:
                p(0) = [0]         
             p(cons) = [0]         
           p(filter) = [0]         
             p(nats) = [0]         
                p(s) = [0]         
            p(sieve) = [2] x1 + [0]
          p(zprimes) = [1]         
        
        Following rules are strictly oriented:
        zprimes() = [1]                   
                  > [0]                   
                  = sieve(nats(s(s(0()))))
        
        
        Following rules are (at-least) weakly oriented:
         filter(cons(X),0(),M) =  [0]       
                               >= [0]       
                               =  cons(0()) 
        
        filter(cons(X),s(N),M) =  [0]       
                               >= [0]       
                               =  cons(X)   
        
                       nats(N) =  [0]       
                               >= [0]       
                               =  cons(N)   
        
              sieve(cons(0())) =  [0]       
                               >= [0]       
                               =  cons(0()) 
        
             sieve(cons(s(N))) =  [0]       
                               >= [0]       
                               =  cons(s(N))
        
* Step 4: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,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(sieve) = {1}
        
        Following symbols are considered usable:
          {filter,nats,sieve,zprimes}
        TcT has computed the following interpretation:
                p(0) = [4]                  
             p(cons) = [0]                  
           p(filter) = [5] x2 + [1] x3 + [0]
             p(nats) = [1] x1 + [1]         
                p(s) = [2]                  
            p(sieve) = [4] x1 + [2]         
          p(zprimes) = [14]                 
        
        Following rules are strictly oriented:
                  nats(N) = [1] N + [1]
                          > [0]        
                          = cons(N)    
        
         sieve(cons(0())) = [2]        
                          > [0]        
                          = cons(0())  
        
        sieve(cons(s(N))) = [2]        
                          > [0]        
                          = cons(s(N)) 
        
        
        Following rules are (at-least) weakly oriented:
         filter(cons(X),0(),M) =  [1] M + [20]          
                               >= [0]                   
                               =  cons(0())             
        
        filter(cons(X),s(N),M) =  [1] M + [10]          
                               >= [0]                   
                               =  cons(X)               
        
                     zprimes() =  [14]                  
                               >= [14]                  
                               =  sieve(nats(s(s(0()))))
        
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))