* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          mark(x){x -> adx(x)} =
            mark(adx(x)) ->^+ a__adx(mark(x))
              = C[mark(x) = mark(x){}]

** Step 1.b:1: WeightGap WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(a__adx) = {1},
            uargs(a__hd) = {1},
            uargs(a__incr) = {1},
            uargs(a__tl) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]         
              p(a__adx) = [1] x1 + [1]
               p(a__hd) = [1] x1 + [0]
             p(a__incr) = [1] x1 + [0]
             p(a__nats) = [0]         
               p(a__tl) = [1] x1 + [0]
            p(a__zeros) = [0]         
                 p(adx) = [0]         
                p(cons) = [0]         
                  p(hd) = [1] x1 + [0]
                p(incr) = [1] x1 + [0]
                p(mark) = [0]         
                p(nats) = [0]         
                   p(s) = [0]         
                  p(tl) = [1] x1 + [0]
               p(zeros) = [0]         
          
          Following rules are strictly oriented:
                  a__adx(X) = [1] X + [1]            
                            > [0]                    
                            = adx(X)                 
          
          a__adx(cons(X,Y)) = [1]                    
                            > [0]                    
                            = a__incr(cons(X,adx(Y)))
          
          
          Following rules are (at-least) weakly oriented:
                    a__hd(X) =  [1] X + [0]       
                             >= [1] X + [0]       
                             =  hd(X)             
          
            a__hd(cons(X,Y)) =  [0]               
                             >= [0]               
                             =  mark(X)           
          
                  a__incr(X) =  [1] X + [0]       
                             >= [1] X + [0]       
                             =  incr(X)           
          
          a__incr(cons(X,Y)) =  [0]               
                             >= [0]               
                             =  cons(s(X),incr(Y))
          
                   a__nats() =  [0]               
                             >= [1]               
                             =  a__adx(a__zeros())
          
                   a__nats() =  [0]               
                             >= [0]               
                             =  nats()            
          
                    a__tl(X) =  [1] X + [0]       
                             >= [1] X + [0]       
                             =  tl(X)             
          
            a__tl(cons(X,Y)) =  [0]               
                             >= [0]               
                             =  mark(Y)           
          
                  a__zeros() =  [0]               
                             >= [0]               
                             =  cons(0(),zeros()) 
          
                  a__zeros() =  [0]               
                             >= [0]               
                             =  zeros()           
          
                   mark(0()) =  [0]               
                             >= [0]               
                             =  0()               
          
                mark(adx(X)) =  [0]               
                             >= [1]               
                             =  a__adx(mark(X))   
          
           mark(cons(X1,X2)) =  [0]               
                             >= [0]               
                             =  cons(X1,X2)       
          
                 mark(hd(X)) =  [0]               
                             >= [0]               
                             =  a__hd(mark(X))    
          
               mark(incr(X)) =  [0]               
                             >= [0]               
                             =  a__incr(mark(X))  
          
                mark(nats()) =  [0]               
                             >= [0]               
                             =  a__nats()         
          
                  mark(s(X)) =  [0]               
                             >= [0]               
                             =  s(X)              
          
                 mark(tl(X)) =  [0]               
                             >= [0]               
                             =  a__tl(mark(X))    
          
               mark(zeros()) =  [0]               
                             >= [0]               
                             =  a__zeros()        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(a__adx) = {1},
            uargs(a__hd) = {1},
            uargs(a__incr) = {1},
            uargs(a__tl) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                  
              p(a__adx) = [1] x1 + [15]        
               p(a__hd) = [1] x1 + [0]         
             p(a__incr) = [1] x1 + [0]         
             p(a__nats) = [0]                  
               p(a__tl) = [1] x1 + [0]         
            p(a__zeros) = [0]                  
                 p(adx) = [1] x1 + [15]        
                p(cons) = [1] x1 + [1] x2 + [9]
                  p(hd) = [1] x1 + [0]         
                p(incr) = [1] x1 + [0]         
                p(mark) = [1] x1 + [0]         
                p(nats) = [0]                  
                   p(s) = [1] x1 + [0]         
                  p(tl) = [1] x1 + [0]         
               p(zeros) = [9]                  
          
          Following rules are strictly oriented:
          a__hd(cons(X,Y)) = [1] X + [1] Y + [9]
                           > [1] X + [0]        
                           = mark(X)            
          
          a__tl(cons(X,Y)) = [1] X + [1] Y + [9]
                           > [1] Y + [0]        
                           = mark(Y)            
          
             mark(zeros()) = [9]                
                           > [0]                
                           = a__zeros()         
          
          
          Following rules are (at-least) weakly oriented:
                   a__adx(X) =  [1] X + [15]           
                             >= [1] X + [15]           
                             =  adx(X)                 
          
           a__adx(cons(X,Y)) =  [1] X + [1] Y + [24]   
                             >= [1] X + [1] Y + [24]   
                             =  a__incr(cons(X,adx(Y)))
          
                    a__hd(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  hd(X)                  
          
                  a__incr(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  incr(X)                
          
          a__incr(cons(X,Y)) =  [1] X + [1] Y + [9]    
                             >= [1] X + [1] Y + [9]    
                             =  cons(s(X),incr(Y))     
          
                   a__nats() =  [0]                    
                             >= [15]                   
                             =  a__adx(a__zeros())     
          
                   a__nats() =  [0]                    
                             >= [0]                    
                             =  nats()                 
          
                    a__tl(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  tl(X)                  
          
                  a__zeros() =  [0]                    
                             >= [18]                   
                             =  cons(0(),zeros())      
          
                  a__zeros() =  [0]                    
                             >= [9]                    
                             =  zeros()                
          
                   mark(0()) =  [0]                    
                             >= [0]                    
                             =  0()                    
          
                mark(adx(X)) =  [1] X + [15]           
                             >= [1] X + [15]           
                             =  a__adx(mark(X))        
          
           mark(cons(X1,X2)) =  [1] X1 + [1] X2 + [9]  
                             >= [1] X1 + [1] X2 + [9]  
                             =  cons(X1,X2)            
          
                 mark(hd(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  a__hd(mark(X))         
          
               mark(incr(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  a__incr(mark(X))       
          
                mark(nats()) =  [0]                    
                             >= [0]                    
                             =  a__nats()              
          
                  mark(s(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  s(X)                   
          
                 mark(tl(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  a__tl(mark(X))         
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__hd(X) -> hd(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(cons(X,Y)) -> mark(X)
            a__tl(cons(X,Y)) -> mark(Y)
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(a__adx) = {1},
            uargs(a__hd) = {1},
            uargs(a__incr) = {1},
            uargs(a__tl) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]         
              p(a__adx) = [1] x1 + [0]
               p(a__hd) = [1] x1 + [6]
             p(a__incr) = [1] x1 + [0]
             p(a__nats) = [0]         
               p(a__tl) = [1] x1 + [0]
            p(a__zeros) = [0]         
                 p(adx) = [1] x1 + [0]
                p(cons) = [0]         
                  p(hd) = [0]         
                p(incr) = [0]         
                p(mark) = [0]         
                p(nats) = [0]         
                   p(s) = [0]         
                  p(tl) = [1] x1 + [0]
               p(zeros) = [0]         
          
          Following rules are strictly oriented:
          a__hd(X) = [1] X + [6]
                   > [0]        
                   = hd(X)      
          
          
          Following rules are (at-least) weakly oriented:
                   a__adx(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  adx(X)                 
          
           a__adx(cons(X,Y)) =  [0]                    
                             >= [0]                    
                             =  a__incr(cons(X,adx(Y)))
          
            a__hd(cons(X,Y)) =  [6]                    
                             >= [0]                    
                             =  mark(X)                
          
                  a__incr(X) =  [1] X + [0]            
                             >= [0]                    
                             =  incr(X)                
          
          a__incr(cons(X,Y)) =  [0]                    
                             >= [0]                    
                             =  cons(s(X),incr(Y))     
          
                   a__nats() =  [0]                    
                             >= [0]                    
                             =  a__adx(a__zeros())     
          
                   a__nats() =  [0]                    
                             >= [0]                    
                             =  nats()                 
          
                    a__tl(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  tl(X)                  
          
            a__tl(cons(X,Y)) =  [0]                    
                             >= [0]                    
                             =  mark(Y)                
          
                  a__zeros() =  [0]                    
                             >= [0]                    
                             =  cons(0(),zeros())      
          
                  a__zeros() =  [0]                    
                             >= [0]                    
                             =  zeros()                
          
                   mark(0()) =  [0]                    
                             >= [0]                    
                             =  0()                    
          
                mark(adx(X)) =  [0]                    
                             >= [0]                    
                             =  a__adx(mark(X))        
          
           mark(cons(X1,X2)) =  [0]                    
                             >= [0]                    
                             =  cons(X1,X2)            
          
                 mark(hd(X)) =  [0]                    
                             >= [6]                    
                             =  a__hd(mark(X))         
          
               mark(incr(X)) =  [0]                    
                             >= [0]                    
                             =  a__incr(mark(X))       
          
                mark(nats()) =  [0]                    
                             >= [0]                    
                             =  a__nats()              
          
                  mark(s(X)) =  [0]                    
                             >= [0]                    
                             =  s(X)                   
          
                 mark(tl(X)) =  [0]                    
                             >= [0]                    
                             =  a__tl(mark(X))         
          
               mark(zeros()) =  [0]                    
                             >= [0]                    
                             =  a__zeros()             
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__tl(cons(X,Y)) -> mark(Y)
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(a__adx) = {1},
            uargs(a__hd) = {1},
            uargs(a__incr) = {1},
            uargs(a__tl) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]         
              p(a__adx) = [1] x1 + [1]
               p(a__hd) = [1] x1 + [0]
             p(a__incr) = [1] x1 + [1]
             p(a__nats) = [0]         
               p(a__tl) = [1] x1 + [0]
            p(a__zeros) = [1]         
                 p(adx) = [0]         
                p(cons) = [1]         
                  p(hd) = [0]         
                p(incr) = [1] x1 + [0]
                p(mark) = [1]         
                p(nats) = [0]         
                   p(s) = [0]         
                  p(tl) = [1] x1 + [0]
               p(zeros) = [1]         
          
          Following rules are strictly oriented:
                  a__incr(X) = [1] X + [1]       
                             > [1] X + [0]       
                             = incr(X)           
          
          a__incr(cons(X,Y)) = [2]               
                             > [1]               
                             = cons(s(X),incr(Y))
          
                   mark(0()) = [1]               
                             > [0]               
                             = 0()               
          
                mark(nats()) = [1]               
                             > [0]               
                             = a__nats()         
          
                  mark(s(X)) = [1]               
                             > [0]               
                             = s(X)              
          
          
          Following rules are (at-least) weakly oriented:
                  a__adx(X) =  [1] X + [1]            
                            >= [0]                    
                            =  adx(X)                 
          
          a__adx(cons(X,Y)) =  [2]                    
                            >= [2]                    
                            =  a__incr(cons(X,adx(Y)))
          
                   a__hd(X) =  [1] X + [0]            
                            >= [0]                    
                            =  hd(X)                  
          
           a__hd(cons(X,Y)) =  [1]                    
                            >= [1]                    
                            =  mark(X)                
          
                  a__nats() =  [0]                    
                            >= [2]                    
                            =  a__adx(a__zeros())     
          
                  a__nats() =  [0]                    
                            >= [0]                    
                            =  nats()                 
          
                   a__tl(X) =  [1] X + [0]            
                            >= [1] X + [0]            
                            =  tl(X)                  
          
           a__tl(cons(X,Y)) =  [1]                    
                            >= [1]                    
                            =  mark(Y)                
          
                 a__zeros() =  [1]                    
                            >= [1]                    
                            =  cons(0(),zeros())      
          
                 a__zeros() =  [1]                    
                            >= [1]                    
                            =  zeros()                
          
               mark(adx(X)) =  [1]                    
                            >= [2]                    
                            =  a__adx(mark(X))        
          
          mark(cons(X1,X2)) =  [1]                    
                            >= [1]                    
                            =  cons(X1,X2)            
          
                mark(hd(X)) =  [1]                    
                            >= [1]                    
                            =  a__hd(mark(X))         
          
              mark(incr(X)) =  [1]                    
                            >= [2]                    
                            =  a__incr(mark(X))       
          
                mark(tl(X)) =  [1]                    
                            >= [1]                    
                            =  a__tl(mark(X))         
          
              mark(zeros()) =  [1]                    
                            >= [1]                    
                            =  a__zeros()             
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(tl(X)) -> a__tl(mark(X))
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__tl(cons(X,Y)) -> mark(Y)
            mark(0()) -> 0()
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(a__adx) = {1},
            uargs(a__hd) = {1},
            uargs(a__incr) = {1},
            uargs(a__tl) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [5]                  
              p(a__adx) = [1] x1 + [0]         
               p(a__hd) = [1] x1 + [0]         
             p(a__incr) = [1] x1 + [0]         
             p(a__nats) = [0]                  
               p(a__tl) = [1] x1 + [0]         
            p(a__zeros) = [0]                  
                 p(adx) = [1] x1 + [0]         
                p(cons) = [1] x1 + [1] x2 + [0]
                  p(hd) = [1] x1 + [0]         
                p(incr) = [1] x1 + [0]         
                p(mark) = [1] x1 + [0]         
                p(nats) = [0]                  
                   p(s) = [1] x1 + [0]         
                  p(tl) = [1] x1 + [1]         
               p(zeros) = [0]                  
          
          Following rules are strictly oriented:
          mark(tl(X)) = [1] X + [1]   
                      > [1] X + [0]   
                      = a__tl(mark(X))
          
          
          Following rules are (at-least) weakly oriented:
                   a__adx(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  adx(X)                 
          
           a__adx(cons(X,Y)) =  [1] X + [1] Y + [0]    
                             >= [1] X + [1] Y + [0]    
                             =  a__incr(cons(X,adx(Y)))
          
                    a__hd(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  hd(X)                  
          
            a__hd(cons(X,Y)) =  [1] X + [1] Y + [0]    
                             >= [1] X + [0]            
                             =  mark(X)                
          
                  a__incr(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  incr(X)                
          
          a__incr(cons(X,Y)) =  [1] X + [1] Y + [0]    
                             >= [1] X + [1] Y + [0]    
                             =  cons(s(X),incr(Y))     
          
                   a__nats() =  [0]                    
                             >= [0]                    
                             =  a__adx(a__zeros())     
          
                   a__nats() =  [0]                    
                             >= [0]                    
                             =  nats()                 
          
                    a__tl(X) =  [1] X + [0]            
                             >= [1] X + [1]            
                             =  tl(X)                  
          
            a__tl(cons(X,Y)) =  [1] X + [1] Y + [0]    
                             >= [1] Y + [0]            
                             =  mark(Y)                
          
                  a__zeros() =  [0]                    
                             >= [5]                    
                             =  cons(0(),zeros())      
          
                  a__zeros() =  [0]                    
                             >= [0]                    
                             =  zeros()                
          
                   mark(0()) =  [5]                    
                             >= [5]                    
                             =  0()                    
          
                mark(adx(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  a__adx(mark(X))        
          
           mark(cons(X1,X2)) =  [1] X1 + [1] X2 + [0]  
                             >= [1] X1 + [1] X2 + [0]  
                             =  cons(X1,X2)            
          
                 mark(hd(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  a__hd(mark(X))         
          
               mark(incr(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  a__incr(mark(X))       
          
                mark(nats()) =  [0]                    
                             >= [0]                    
                             =  a__nats()              
          
                  mark(s(X)) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  s(X)                   
          
               mark(zeros()) =  [0]                    
                             >= [0]                    
                             =  a__zeros()             
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__tl(cons(X,Y)) -> mark(Y)
            mark(0()) -> 0()
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(a__adx) = {1},
            uargs(a__hd) = {1},
            uargs(a__incr) = {1},
            uargs(a__tl) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                  
              p(a__adx) = [1] x1 + [0]         
               p(a__hd) = [1] x1 + [4]         
             p(a__incr) = [1] x1 + [0]         
             p(a__nats) = [5]                  
               p(a__tl) = [1] x1 + [4]         
            p(a__zeros) = [3]                  
                 p(adx) = [1] x1 + [0]         
                p(cons) = [1] x1 + [1] x2 + [1]
                  p(hd) = [1] x1 + [4]         
                p(incr) = [1] x1 + [0]         
                p(mark) = [1] x1 + [5]         
                p(nats) = [0]                  
                   p(s) = [1] x1 + [0]         
                  p(tl) = [1] x1 + [4]         
               p(zeros) = [0]                  
          
          Following rules are strictly oriented:
                  a__nats() = [5]                  
                            > [3]                  
                            = a__adx(a__zeros())   
          
                  a__nats() = [5]                  
                            > [0]                  
                            = nats()               
          
                 a__zeros() = [3]                  
                            > [1]                  
                            = cons(0(),zeros())    
          
                 a__zeros() = [3]                  
                            > [0]                  
                            = zeros()              
          
          mark(cons(X1,X2)) = [1] X1 + [1] X2 + [6]
                            > [1] X1 + [1] X2 + [1]
                            = cons(X1,X2)          
          
          
          Following rules are (at-least) weakly oriented:
                   a__adx(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  adx(X)                 
          
           a__adx(cons(X,Y)) =  [1] X + [1] Y + [1]    
                             >= [1] X + [1] Y + [1]    
                             =  a__incr(cons(X,adx(Y)))
          
                    a__hd(X) =  [1] X + [4]            
                             >= [1] X + [4]            
                             =  hd(X)                  
          
            a__hd(cons(X,Y)) =  [1] X + [1] Y + [5]    
                             >= [1] X + [5]            
                             =  mark(X)                
          
                  a__incr(X) =  [1] X + [0]            
                             >= [1] X + [0]            
                             =  incr(X)                
          
          a__incr(cons(X,Y)) =  [1] X + [1] Y + [1]    
                             >= [1] X + [1] Y + [1]    
                             =  cons(s(X),incr(Y))     
          
                    a__tl(X) =  [1] X + [4]            
                             >= [1] X + [4]            
                             =  tl(X)                  
          
            a__tl(cons(X,Y)) =  [1] X + [1] Y + [5]    
                             >= [1] Y + [5]            
                             =  mark(Y)                
          
                   mark(0()) =  [5]                    
                             >= [0]                    
                             =  0()                    
          
                mark(adx(X)) =  [1] X + [5]            
                             >= [1] X + [5]            
                             =  a__adx(mark(X))        
          
                 mark(hd(X)) =  [1] X + [9]            
                             >= [1] X + [9]            
                             =  a__hd(mark(X))         
          
               mark(incr(X)) =  [1] X + [5]            
                             >= [1] X + [5]            
                             =  a__incr(mark(X))       
          
                mark(nats()) =  [5]                    
                             >= [5]                    
                             =  a__nats()              
          
                  mark(s(X)) =  [1] X + [5]            
                             >= [1] X + [0]            
                             =  s(X)                   
          
                 mark(tl(X)) =  [1] X + [9]            
                             >= [1] X + [9]            
                             =  a__tl(mark(X))         
          
               mark(zeros()) =  [5]                    
                             >= [3]                    
                             =  a__zeros()             
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: MI WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__tl(X) -> tl(X)
            mark(adx(X)) -> a__adx(mark(X))
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 1))), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 1))):
        
        The following argument positions are considered usable:
          uargs(a__adx) = {1},
          uargs(a__hd) = {1},
          uargs(a__incr) = {1},
          uargs(a__tl) = {1}
        
        Following symbols are considered usable:
          {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark}
        TcT has computed the following interpretation:
                 p(0) = [0]                        
                        [0]                        
            p(a__adx) = [1 0] x_1 + [0]            
                        [0 1]       [0]            
             p(a__hd) = [1 0] x_1 + [7]            
                        [0 1]       [4]            
           p(a__incr) = [1 0] x_1 + [0]            
                        [0 1]       [0]            
           p(a__nats) = [1]                        
                        [3]                        
             p(a__tl) = [1 0] x_1 + [3]            
                        [0 1]       [2]            
          p(a__zeros) = [1]                        
                        [0]                        
               p(adx) = [0 0] x_1 + [0]            
                        [0 1]       [0]            
              p(cons) = [0 2] x_1 + [0 2] x_2 + [0]
                        [0 1]       [0 1]       [0]
                p(hd) = [0 0] x_1 + [0]            
                        [0 1]       [4]            
              p(incr) = [0 0] x_1 + [0]            
                        [0 1]       [0]            
              p(mark) = [0 2] x_1 + [1]            
                        [0 1]       [0]            
              p(nats) = [1]                        
                        [3]                        
                 p(s) = [1]                        
                        [0]                        
                p(tl) = [0 0] x_1 + [0]            
                        [0 1]       [2]            
             p(zeros) = [0]                        
                        [0]                        
        
        Following rules are strictly oriented:
           a__tl(X) = [1 0] X + [3] 
                      [0 1]     [2] 
                    > [0 0] X + [0] 
                      [0 1]     [2] 
                    = tl(X)         
        
        mark(hd(X)) = [0 2] X + [9] 
                      [0 1]     [4] 
                    > [0 2] X + [8] 
                      [0 1]     [4] 
                    = a__hd(mark(X))
        
        
        Following rules are (at-least) weakly oriented:
                 a__adx(X) =  [1 0] X + [0]            
                              [0 1]     [0]            
                           >= [0 0] X + [0]            
                              [0 1]     [0]            
                           =  adx(X)                   
        
         a__adx(cons(X,Y)) =  [0 2] X + [0 2] Y + [0]  
                              [0 1]     [0 1]     [0]  
                           >= [0 2] X + [0 2] Y + [0]  
                              [0 1]     [0 1]     [0]  
                           =  a__incr(cons(X,adx(Y)))  
        
                  a__hd(X) =  [1 0] X + [7]            
                              [0 1]     [4]            
                           >= [0 0] X + [0]            
                              [0 1]     [4]            
                           =  hd(X)                    
        
          a__hd(cons(X,Y)) =  [0 2] X + [0 2] Y + [7]  
                              [0 1]     [0 1]     [4]  
                           >= [0 2] X + [1]            
                              [0 1]     [0]            
                           =  mark(X)                  
        
                a__incr(X) =  [1 0] X + [0]            
                              [0 1]     [0]            
                           >= [0 0] X + [0]            
                              [0 1]     [0]            
                           =  incr(X)                  
        
        a__incr(cons(X,Y)) =  [0 2] X + [0 2] Y + [0]  
                              [0 1]     [0 1]     [0]  
                           >= [0 2] Y + [0]            
                              [0 1]     [0]            
                           =  cons(s(X),incr(Y))       
        
                 a__nats() =  [1]                      
                              [3]                      
                           >= [1]                      
                              [0]                      
                           =  a__adx(a__zeros())       
        
                 a__nats() =  [1]                      
                              [3]                      
                           >= [1]                      
                              [3]                      
                           =  nats()                   
        
          a__tl(cons(X,Y)) =  [0 2] X + [0 2] Y + [3]  
                              [0 1]     [0 1]     [2]  
                           >= [0 2] Y + [1]            
                              [0 1]     [0]            
                           =  mark(Y)                  
        
                a__zeros() =  [1]                      
                              [0]                      
                           >= [0]                      
                              [0]                      
                           =  cons(0(),zeros())        
        
                a__zeros() =  [1]                      
                              [0]                      
                           >= [0]                      
                              [0]                      
                           =  zeros()                  
        
                 mark(0()) =  [1]                      
                              [0]                      
                           >= [0]                      
                              [0]                      
                           =  0()                      
        
              mark(adx(X)) =  [0 2] X + [1]            
                              [0 1]     [0]            
                           >= [0 2] X + [1]            
                              [0 1]     [0]            
                           =  a__adx(mark(X))          
        
         mark(cons(X1,X2)) =  [0 2] X1 + [0 2] X2 + [1]
                              [0 1]      [0 1]      [0]
                           >= [0 2] X1 + [0 2] X2 + [0]
                              [0 1]      [0 1]      [0]
                           =  cons(X1,X2)              
        
             mark(incr(X)) =  [0 2] X + [1]            
                              [0 1]     [0]            
                           >= [0 2] X + [1]            
                              [0 1]     [0]            
                           =  a__incr(mark(X))         
        
              mark(nats()) =  [7]                      
                              [3]                      
                           >= [1]                      
                              [3]                      
                           =  a__nats()                
        
                mark(s(X)) =  [1]                      
                              [0]                      
                           >= [1]                      
                              [0]                      
                           =  s(X)                     
        
               mark(tl(X)) =  [0 2] X + [5]            
                              [0 1]     [2]            
                           >= [0 2] X + [4]            
                              [0 1]     [2]            
                           =  a__tl(mark(X))           
        
             mark(zeros()) =  [1]                      
                              [0]                      
                           >= [1]                      
                              [0]                      
                           =  a__zeros()               
        
** Step 1.b:8: MI WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            mark(adx(X)) -> a__adx(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(a__adx) = {1},
          uargs(a__hd) = {1},
          uargs(a__incr) = {1},
          uargs(a__tl) = {1}
        
        Following symbols are considered usable:
          {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark}
        TcT has computed the following interpretation:
                 p(0) = [0]                        
                        [0]                        
            p(a__adx) = [1 0] x_1 + [0]            
                        [0 1]       [2]            
             p(a__hd) = [1 4] x_1 + [6]            
                        [0 1]       [2]            
           p(a__incr) = [1 0] x_1 + [0]            
                        [0 1]       [0]            
           p(a__nats) = [0]                        
                        [2]                        
             p(a__tl) = [1 4] x_1 + [2]            
                        [0 1]       [2]            
          p(a__zeros) = [0]                        
                        [0]                        
               p(adx) = [1 0] x_1 + [0]            
                        [0 1]       [2]            
              p(cons) = [1 0] x_1 + [1 0] x_2 + [0]
                        [0 1]       [0 1]       [0]
                p(hd) = [1 4] x_1 + [4]            
                        [0 1]       [2]            
              p(incr) = [1 0] x_1 + [0]            
                        [0 1]       [0]            
              p(mark) = [1 4] x_1 + [1]            
                        [0 1]       [0]            
              p(nats) = [0]                        
                        [2]                        
                 p(s) = [0 0] x_1 + [0]            
                        [0 1]       [0]            
                p(tl) = [1 4] x_1 + [2]            
                        [0 1]       [2]            
             p(zeros) = [0]                        
                        [0]                        
        
        Following rules are strictly oriented:
        mark(adx(X)) = [1 4] X + [9]  
                       [0 1]     [2]  
                     > [1 4] X + [1]  
                       [0 1]     [2]  
                     = a__adx(mark(X))
        
        
        Following rules are (at-least) weakly oriented:
                 a__adx(X) =  [1 0] X + [0]            
                              [0 1]     [2]            
                           >= [1 0] X + [0]            
                              [0 1]     [2]            
                           =  adx(X)                   
        
         a__adx(cons(X,Y)) =  [1 0] X + [1 0] Y + [0]  
                              [0 1]     [0 1]     [2]  
                           >= [1 0] X + [1 0] Y + [0]  
                              [0 1]     [0 1]     [2]  
                           =  a__incr(cons(X,adx(Y)))  
        
                  a__hd(X) =  [1 4] X + [6]            
                              [0 1]     [2]            
                           >= [1 4] X + [4]            
                              [0 1]     [2]            
                           =  hd(X)                    
        
          a__hd(cons(X,Y)) =  [1 4] X + [1 4] Y + [6]  
                              [0 1]     [0 1]     [2]  
                           >= [1 4] X + [1]            
                              [0 1]     [0]            
                           =  mark(X)                  
        
                a__incr(X) =  [1 0] X + [0]            
                              [0 1]     [0]            
                           >= [1 0] X + [0]            
                              [0 1]     [0]            
                           =  incr(X)                  
        
        a__incr(cons(X,Y)) =  [1 0] X + [1 0] Y + [0]  
                              [0 1]     [0 1]     [0]  
                           >= [0 0] X + [1 0] Y + [0]  
                              [0 1]     [0 1]     [0]  
                           =  cons(s(X),incr(Y))       
        
                 a__nats() =  [0]                      
                              [2]                      
                           >= [0]                      
                              [2]                      
                           =  a__adx(a__zeros())       
        
                 a__nats() =  [0]                      
                              [2]                      
                           >= [0]                      
                              [2]                      
                           =  nats()                   
        
                  a__tl(X) =  [1 4] X + [2]            
                              [0 1]     [2]            
                           >= [1 4] X + [2]            
                              [0 1]     [2]            
                           =  tl(X)                    
        
          a__tl(cons(X,Y)) =  [1 4] X + [1 4] Y + [2]  
                              [0 1]     [0 1]     [2]  
                           >= [1 4] Y + [1]            
                              [0 1]     [0]            
                           =  mark(Y)                  
        
                a__zeros() =  [0]                      
                              [0]                      
                           >= [0]                      
                              [0]                      
                           =  cons(0(),zeros())        
        
                a__zeros() =  [0]                      
                              [0]                      
                           >= [0]                      
                              [0]                      
                           =  zeros()                  
        
                 mark(0()) =  [1]                      
                              [0]                      
                           >= [0]                      
                              [0]                      
                           =  0()                      
        
         mark(cons(X1,X2)) =  [1 4] X1 + [1 4] X2 + [1]
                              [0 1]      [0 1]      [0]
                           >= [1 0] X1 + [1 0] X2 + [0]
                              [0 1]      [0 1]      [0]
                           =  cons(X1,X2)              
        
               mark(hd(X)) =  [1 8] X + [13]           
                              [0 1]     [2]            
                           >= [1 8] X + [7]            
                              [0 1]     [2]            
                           =  a__hd(mark(X))           
        
             mark(incr(X)) =  [1 4] X + [1]            
                              [0 1]     [0]            
                           >= [1 4] X + [1]            
                              [0 1]     [0]            
                           =  a__incr(mark(X))         
        
              mark(nats()) =  [9]                      
                              [2]                      
                           >= [0]                      
                              [2]                      
                           =  a__nats()                
        
                mark(s(X)) =  [0 4] X + [1]            
                              [0 1]     [0]            
                           >= [0 0] X + [0]            
                              [0 1]     [0]            
                           =  s(X)                     
        
               mark(tl(X)) =  [1 8] X + [11]           
                              [0 1]     [2]            
                           >= [1 8] X + [3]            
                              [0 1]     [2]            
                           =  a__tl(mark(X))           
        
             mark(zeros()) =  [1]                      
                              [0]                      
                           >= [0]                      
                              [0]                      
                           =  a__zeros()               
        
** Step 1.b:9: MI WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            mark(incr(X)) -> a__incr(mark(X))
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):
        
        The following argument positions are considered usable:
          uargs(a__adx) = {1},
          uargs(a__hd) = {1},
          uargs(a__incr) = {1},
          uargs(a__tl) = {1}
        
        Following symbols are considered usable:
          {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark}
        TcT has computed the following interpretation:
                 p(0) = [0]                                
                        [0]                                
                        [0]                                
                        [0]                                
            p(a__adx) = [1 0 0 2]       [2]                
                        [0 1 0 2] x_1 + [2]                
                        [0 0 1 1]       [3]                
                        [0 0 0 1]       [2]                
             p(a__hd) = [1 0 1 0]       [0]                
                        [0 1 1 0] x_1 + [1]                
                        [0 0 1 2]       [3]                
                        [0 0 0 1]       [0]                
           p(a__incr) = [1 0 0 1]       [0]                
                        [0 1 0 1] x_1 + [0]                
                        [0 0 1 0]       [1]                
                        [0 0 0 1]       [0]                
           p(a__nats) = [2]                                
                        [3]                                
                        [3]                                
                        [2]                                
             p(a__tl) = [1 0 1 0]       [2]                
                        [0 1 1 0] x_1 + [2]                
                        [0 0 1 2]       [3]                
                        [0 0 0 1]       [0]                
          p(a__zeros) = [0]                                
                        [0]                                
                        [0]                                
                        [0]                                
               p(adx) = [0 0 0 0]       [0]                
                        [0 1 0 1] x_1 + [0]                
                        [0 0 1 1]       [2]                
                        [0 0 0 1]       [2]                
              p(cons) = [0 1 0 0]       [0 1 0 0]       [0]
                        [0 1 0 0] x_1 + [0 1 0 0] x_2 + [0]
                        [0 0 1 0]       [0 0 1 0]       [0]
                        [0 0 0 1]       [0 0 0 1]       [0]
                p(hd) = [0 0 1 0]       [0]                
                        [0 1 1 0] x_1 + [1]                
                        [0 0 1 2]       [3]                
                        [0 0 0 1]       [0]                
              p(incr) = [0 0 0 0]       [0]                
                        [0 1 0 1] x_1 + [0]                
                        [0 0 1 0]       [1]                
                        [0 0 0 1]       [0]                
              p(mark) = [0 1 1 0]       [0]                
                        [0 1 1 0] x_1 + [0]                
                        [0 0 1 2]       [0]                
                        [0 0 0 1]       [0]                
              p(nats) = [0]                                
                        [3]                                
                        [0]                                
                        [2]                                
                 p(s) = [0 0 0 0]       [0]                
                        [0 0 0 0] x_1 + [0]                
                        [0 0 0 0]       [0]                
                        [0 0 0 1]       [0]                
                p(tl) = [0 0 0 0]       [2]                
                        [0 1 1 0] x_1 + [1]                
                        [0 0 1 2]       [3]                
                        [0 0 0 1]       [0]                
             p(zeros) = [0]                                
                        [0]                                
                        [0]                                
                        [0]                                
        
        Following rules are strictly oriented:
        mark(incr(X)) = [0 1 1 1]     [1]
                        [0 1 1 1] X + [1]
                        [0 0 1 2]     [1]
                        [0 0 0 1]     [0]
                      > [0 1 1 1]     [0]
                        [0 1 1 1] X + [0]
                        [0 0 1 2]     [1]
                        [0 0 0 1]     [0]
                      = a__incr(mark(X)) 
        
        
        Following rules are (at-least) weakly oriented:
                 a__adx(X) =  [1 0 0 2]     [2]                
                              [0 1 0 2] X + [2]                
                              [0 0 1 1]     [3]                
                              [0 0 0 1]     [2]                
                           >= [0 0 0 0]     [0]                
                              [0 1 0 1] X + [0]                
                              [0 0 1 1]     [2]                
                              [0 0 0 1]     [2]                
                           =  adx(X)                           
        
         a__adx(cons(X,Y)) =  [0 1 0 2]     [0 1 0 2]     [2]  
                              [0 1 0 2] X + [0 1 0 2] Y + [2]  
                              [0 0 1 1]     [0 0 1 1]     [3]  
                              [0 0 0 1]     [0 0 0 1]     [2]  
                           >= [0 1 0 1]     [0 1 0 2]     [2]  
                              [0 1 0 1] X + [0 1 0 2] Y + [2]  
                              [0 0 1 0]     [0 0 1 1]     [3]  
                              [0 0 0 1]     [0 0 0 1]     [2]  
                           =  a__incr(cons(X,adx(Y)))          
        
                  a__hd(X) =  [1 0 1 0]     [0]                
                              [0 1 1 0] X + [1]                
                              [0 0 1 2]     [3]                
                              [0 0 0 1]     [0]                
                           >= [0 0 1 0]     [0]                
                              [0 1 1 0] X + [1]                
                              [0 0 1 2]     [3]                
                              [0 0 0 1]     [0]                
                           =  hd(X)                            
        
          a__hd(cons(X,Y)) =  [0 1 1 0]     [0 1 1 0]     [0]  
                              [0 1 1 0] X + [0 1 1 0] Y + [1]  
                              [0 0 1 2]     [0 0 1 2]     [3]  
                              [0 0 0 1]     [0 0 0 1]     [0]  
                           >= [0 1 1 0]     [0]                
                              [0 1 1 0] X + [0]                
                              [0 0 1 2]     [0]                
                              [0 0 0 1]     [0]                
                           =  mark(X)                          
        
                a__incr(X) =  [1 0 0 1]     [0]                
                              [0 1 0 1] X + [0]                
                              [0 0 1 0]     [1]                
                              [0 0 0 1]     [0]                
                           >= [0 0 0 0]     [0]                
                              [0 1 0 1] X + [0]                
                              [0 0 1 0]     [1]                
                              [0 0 0 1]     [0]                
                           =  incr(X)                          
        
        a__incr(cons(X,Y)) =  [0 1 0 1]     [0 1 0 1]     [0]  
                              [0 1 0 1] X + [0 1 0 1] Y + [0]  
                              [0 0 1 0]     [0 0 1 0]     [1]  
                              [0 0 0 1]     [0 0 0 1]     [0]  
                           >= [0 0 0 0]     [0 1 0 1]     [0]  
                              [0 0 0 0] X + [0 1 0 1] Y + [0]  
                              [0 0 0 0]     [0 0 1 0]     [1]  
                              [0 0 0 1]     [0 0 0 1]     [0]  
                           =  cons(s(X),incr(Y))               
        
                 a__nats() =  [2]                              
                              [3]                              
                              [3]                              
                              [2]                              
                           >= [2]                              
                              [2]                              
                              [3]                              
                              [2]                              
                           =  a__adx(a__zeros())               
        
                 a__nats() =  [2]                              
                              [3]                              
                              [3]                              
                              [2]                              
                           >= [0]                              
                              [3]                              
                              [0]                              
                              [2]                              
                           =  nats()                           
        
                  a__tl(X) =  [1 0 1 0]     [2]                
                              [0 1 1 0] X + [2]                
                              [0 0 1 2]     [3]                
                              [0 0 0 1]     [0]                
                           >= [0 0 0 0]     [2]                
                              [0 1 1 0] X + [1]                
                              [0 0 1 2]     [3]                
                              [0 0 0 1]     [0]                
                           =  tl(X)                            
        
          a__tl(cons(X,Y)) =  [0 1 1 0]     [0 1 1 0]     [2]  
                              [0 1 1 0] X + [0 1 1 0] Y + [2]  
                              [0 0 1 2]     [0 0 1 2]     [3]  
                              [0 0 0 1]     [0 0 0 1]     [0]  
                           >= [0 1 1 0]     [0]                
                              [0 1 1 0] Y + [0]                
                              [0 0 1 2]     [0]                
                              [0 0 0 1]     [0]                
                           =  mark(Y)                          
        
                a__zeros() =  [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           >= [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           =  cons(0(),zeros())                
        
                a__zeros() =  [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           >= [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           =  zeros()                          
        
                 mark(0()) =  [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           >= [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           =  0()                              
        
              mark(adx(X)) =  [0 1 1 2]     [2]                
                              [0 1 1 2] X + [2]                
                              [0 0 1 3]     [6]                
                              [0 0 0 1]     [2]                
                           >= [0 1 1 2]     [2]                
                              [0 1 1 2] X + [2]                
                              [0 0 1 3]     [3]                
                              [0 0 0 1]     [2]                
                           =  a__adx(mark(X))                  
        
         mark(cons(X1,X2)) =  [0 1 1 0]      [0 1 1 0]      [0]
                              [0 1 1 0] X1 + [0 1 1 0] X2 + [0]
                              [0 0 1 2]      [0 0 1 2]      [0]
                              [0 0 0 1]      [0 0 0 1]      [0]
                           >= [0 1 0 0]      [0 1 0 0]      [0]
                              [0 1 0 0] X1 + [0 1 0 0] X2 + [0]
                              [0 0 1 0]      [0 0 1 0]      [0]
                              [0 0 0 1]      [0 0 0 1]      [0]
                           =  cons(X1,X2)                      
        
               mark(hd(X)) =  [0 1 2 2]     [4]                
                              [0 1 2 2] X + [4]                
                              [0 0 1 4]     [3]                
                              [0 0 0 1]     [0]                
                           >= [0 1 2 2]     [0]                
                              [0 1 2 2] X + [1]                
                              [0 0 1 4]     [3]                
                              [0 0 0 1]     [0]                
                           =  a__hd(mark(X))                   
        
              mark(nats()) =  [3]                              
                              [3]                              
                              [4]                              
                              [2]                              
                           >= [2]                              
                              [3]                              
                              [3]                              
                              [2]                              
                           =  a__nats()                        
        
                mark(s(X)) =  [0 0 0 0]     [0]                
                              [0 0 0 0] X + [0]                
                              [0 0 0 2]     [0]                
                              [0 0 0 1]     [0]                
                           >= [0 0 0 0]     [0]                
                              [0 0 0 0] X + [0]                
                              [0 0 0 0]     [0]                
                              [0 0 0 1]     [0]                
                           =  s(X)                             
        
               mark(tl(X)) =  [0 1 2 2]     [4]                
                              [0 1 2 2] X + [4]                
                              [0 0 1 4]     [3]                
                              [0 0 0 1]     [0]                
                           >= [0 1 2 2]     [2]                
                              [0 1 2 2] X + [2]                
                              [0 0 1 4]     [3]                
                              [0 0 0 1]     [0]                
                           =  a__tl(mark(X))                   
        
             mark(zeros()) =  [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           >= [0]                              
                              [0]                              
                              [0]                              
                              [0]                              
                           =  a__zeros()                       
        
** Step 1.b:10: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__adx(X) -> adx(X)
            a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
            a__hd(X) -> hd(X)
            a__hd(cons(X,Y)) -> mark(X)
            a__incr(X) -> incr(X)
            a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
            a__nats() -> a__adx(a__zeros())
            a__nats() -> nats()
            a__tl(X) -> tl(X)
            a__tl(cons(X,Y)) -> mark(Y)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(adx(X)) -> a__adx(mark(X))
            mark(cons(X1,X2)) -> cons(X1,X2)
            mark(hd(X)) -> a__hd(mark(X))
            mark(incr(X)) -> a__incr(mark(X))
            mark(nats()) -> a__nats()
            mark(s(X)) -> s(X)
            mark(tl(X)) -> a__tl(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
            ,tl/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
            ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))