* Step 1: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            cons(x,y) -> x
            cons(x,y) -> y
            f(s(a()),s(b()),x) -> f(x,x,x)
            g(f(s(x),s(y),z)) -> g(f(x,y,z))
        - Signature:
            {cons/2,f/3,g/1} / {a/0,b/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons,f,g} and constructors {a,b,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          cons#(x,y) -> c_1(x)
          cons#(x,y) -> c_2(y)
          f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
          g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
            g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
        - Strict TRS:
            cons(x,y) -> x
            cons(x,y) -> y
            f(s(a()),s(b()),x) -> f(x,x,x)
            g(f(s(x),s(y),z)) -> g(f(x,y,z))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          cons#(x,y) -> c_1(x)
          cons#(x,y) -> c_2(y)
          f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
* Step 3: WeightGap WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            none
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(a) = [0] 
                p(b) = [0] 
             p(cons) = [0] 
                p(f) = [0] 
                p(g) = [0] 
                p(s) = [0] 
            p(cons#) = [0] 
               p(f#) = [15]
               p(g#) = [0] 
              p(c_1) = [0] 
              p(c_2) = [0] 
              p(c_3) = [0] 
              p(c_4) = [0] 
          
          Following rules are strictly oriented:
          f#(s(a()),s(b()),x) = [15]          
                              > [0]           
                              = c_3(f#(x,x,x))
          
          
          Following rules are (at-least) weakly oriented:
          cons#(x,y) =  [0]   
                     >= [0]   
                     =  c_1(x)
          
          cons#(x,y) =  [0]   
                     >= [0]   
                     =  c_2(y)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
        - Weak DPs:
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            none
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(a) = [0] 
                p(b) = [0] 
             p(cons) = [0] 
                p(f) = [0] 
                p(g) = [0] 
                p(s) = [11]
            p(cons#) = [1] 
               p(f#) = [0] 
               p(g#) = [0] 
              p(c_1) = [0] 
              p(c_2) = [0] 
              p(c_3) = [0] 
              p(c_4) = [0] 
          
          Following rules are strictly oriented:
          cons#(x,y) = [1]   
                     > [0]   
                     = c_1(x)
          
          cons#(x,y) = [1]   
                     > [0]   
                     = c_2(y)
          
          
          Following rules are (at-least) weakly oriented:
          f#(s(a()),s(b()),x) =  [0]           
                              >= [0]           
                              =  c_3(f#(x,x,x))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))