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

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
        - Signature:
            {f/3} / {0/0,1/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {0,1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Weak TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          f#(0(),1(),x) -> c_2(f#(s(x),x,x))
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,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(c_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {f#}
        TcT has computed the following interpretation:
            p(0) = 0     
            p(1) = 0     
            p(f) = 0     
            p(s) = 2 + x1
           p(f#) = x3    
          p(c_1) = x1    
          p(c_2) = x1    
        
        Following rules are strictly oriented:
        f#(x,y,s(z)) = 2 + z             
                     > z                 
                     = c_1(f#(0(),1(),z))
        
        
        Following rules are (at-least) weakly oriented:
        f#(0(),1(),x) =  x                
                      >= x                
                      =  c_2(f#(s(x),x,x))
        
** Step 1.b:4: MI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        MI {miKind = Automaton Nothing, miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind Automaton Nothing:
        
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {f#}
        TcT has computed the following interpretation:
            p(0) = [1]                                     
                   [8]                                     
            p(1) = [1]                                     
                   [0]                                     
            p(f) = [2 0] x_1 + [8 8] x_2 + [0]             
                   [2 1]       [0 1]       [0]             
            p(s) = [1 0] x_1 + [8]                         
                   [0 0]       [2]                         
           p(f#) = [0 1] x_1 + [0  0] x_2 + [2 0] x_3 + [1]
                   [0 0]       [8 12]       [0 0]       [0]
          p(c_1) = [1 1] x_1 + [0]                         
                   [0 0]       [0]                         
          p(c_2) = [1 0] x_1 + [0]                         
                   [0 0]       [8]                         
        
        Following rules are strictly oriented:
        f#(0(),1(),x) = [2 0] x + [9]    
                        [0 0]     [8]    
                      > [2 0] x + [3]    
                        [0 0]     [8]    
                      = c_2(f#(s(x),x,x))
        
        
        Following rules are (at-least) weakly oriented:
        f#(x,y,s(z)) =  [0 1] x + [0  0] y + [2 0] z + [17]
                        [0 0]     [8 12]     [0 0]     [0] 
                     >= [2 0] z + [17]                     
                        [0 0]     [0]                      
                     =  c_1(f#(0(),1(),z))                 
        
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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