* Step 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:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        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 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))
        - Strict 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:
             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 3: PredecessorEstimationCP 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:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          
        Consider the set of all dependency pairs
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 3.a:1: NaturalMI 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:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(0) = [0]                  
            p(1) = [1]                  
            p(f) = [1] x2 + [1] x3 + [2]
            p(s) = [1] x1 + [4]         
           p(f#) = [4] x3 + [0]         
          p(c_1) = [1] x1 + [15]        
          p(c_2) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        f#(x,y,s(z)) = [4] z + [16]      
                     > [4] z + [15]      
                     = c_1(f#(0(),1(),z))
        
        
        Following rules are (at-least) weakly oriented:
        f#(0(),1(),x) =  [4] x + [0]      
                      >= [4] x + [0]      
                      =  c_2(f#(s(x),x,x))
        
** Step 3.a:2: Assumption WORST_CASE(?,O(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:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 3.b:1: RemoveWeakSuffixes 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:
             runtime complexity wrt. defined symbols {f#} and constructors {0,1,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
             -->_1 f#(0(),1(),x) -> c_2(f#(s(x),x,x)):2
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
          2:W:f#(0(),1(),x) -> c_2(f#(s(x),x,x))
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
** Step 3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/3,f#/3} / {0/0,1/0,s/1,c_1/1,c_2/1}
        - Obligation:
             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(?,O(n^1))