*** 1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        app(add(n,x),y) -> add(n,app(x,y))
        app(nil(),y) -> y
        reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
        reverse(nil()) -> nil()
        shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
        shuffle(nil()) -> nil()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {app/2,reverse/1,shuffle/1} / {add/2,nil/0}
      Obligation:
        Innermost
        basic terms: {app,reverse,shuffle}/{add,nil}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        app#(add(n,x),y) -> c_1(app#(x,y))
        app#(nil(),y) -> c_2()
        reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        reverse#(nil()) -> c_4()
        shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        shuffle#(nil()) -> c_6()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(add(n,x),y) -> c_1(app#(x,y))
        app#(nil(),y) -> c_2()
        reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        reverse#(nil()) -> c_4()
        shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        shuffle#(nil()) -> c_6()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        app(add(n,x),y) -> add(n,app(x,y))
        app(nil(),y) -> y
        reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
        reverse(nil()) -> nil()
        shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
        shuffle(nil()) -> nil()
      Signature:
        {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
      Obligation:
        Innermost
        basic terms: {app#,reverse#,shuffle#}/{add,nil}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        app(add(n,x),y) -> add(n,app(x,y))
        app(nil(),y) -> y
        reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
        reverse(nil()) -> nil()
        app#(add(n,x),y) -> c_1(app#(x,y))
        app#(nil(),y) -> c_2()
        reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        reverse#(nil()) -> c_4()
        shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        shuffle#(nil()) -> c_6()
*** 1.1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(add(n,x),y) -> c_1(app#(x,y))
        app#(nil(),y) -> c_2()
        reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        reverse#(nil()) -> c_4()
        shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        shuffle#(nil()) -> c_6()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        app(add(n,x),y) -> add(n,app(x,y))
        app(nil(),y) -> y
        reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
        reverse(nil()) -> nil()
      Signature:
        {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
      Obligation:
        Innermost
        basic terms: {app#,reverse#,shuffle#}/{add,nil}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {2,4,6}
      by application of
        Pre({2,4,6}) = {1,3,5}.
      Here rules are labelled as follows:
        1: app#(add(n,x),y) -> c_1(app#(x  
                                       ,y))
        2: app#(nil(),y) -> c_2()          
        3: reverse#(add(n,x)) ->           
             c_3(app#(reverse(x)           
                     ,add(n,nil()))        
                ,reverse#(x))              
        4: reverse#(nil()) -> c_4()        
        5: shuffle#(add(n,x)) ->           
             c_5(shuffle#(reverse(x))      
                ,reverse#(x))              
        6: shuffle#(nil()) -> c_6()        
*** 1.1.1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(add(n,x),y) -> c_1(app#(x,y))
        reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        app#(nil(),y) -> c_2()
        reverse#(nil()) -> c_4()
        shuffle#(nil()) -> c_6()
      Weak TRS Rules:
        app(add(n,x),y) -> add(n,app(x,y))
        app(nil(),y) -> y
        reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
        reverse(nil()) -> nil()
      Signature:
        {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
      Obligation:
        Innermost
        basic terms: {app#,reverse#,shuffle#}/{add,nil}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:app#(add(n,x),y) -> c_1(app#(x,y))
           -->_1 app#(nil(),y) -> c_2():4
           -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
        
        2:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
           -->_2 reverse#(nil()) -> c_4():5
           -->_1 app#(nil(),y) -> c_2():4
           -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
           -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
        
        3:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
           -->_1 shuffle#(nil()) -> c_6():6
           -->_2 reverse#(nil()) -> c_4():5
           -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):3
           -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
        
        4:W:app#(nil(),y) -> c_2()
           
        
        5:W:reverse#(nil()) -> c_4()
           
        
        6:W:shuffle#(nil()) -> c_6()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        6: shuffle#(nil()) -> c_6()
        5: reverse#(nil()) -> c_4()
        4: app#(nil(),y) -> c_2()  
*** 1.1.1.1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(add(n,x),y) -> c_1(app#(x,y))
        reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        app(add(n,x),y) -> add(n,app(x,y))
        app(nil(),y) -> y
        reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
        reverse(nil()) -> nil()
      Signature:
        {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
      Obligation:
        Innermost
        basic terms: {app#,reverse#,shuffle#}/{add,nil}
    Applied Processor:
      Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    Proof:
      We analyse the complexity of following sub-problems (R) and (S).
      Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
      
      Problem (R)
        Strict DP Rules:
          app#(add(n,x),y) -> c_1(app#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        Weak TRS Rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
        Signature:
          {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        Obligation:
          Innermost
          basic terms: {app#,reverse#,shuffle#}/{add,nil}
      
      Problem (S)
        Strict DP Rules:
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          app#(add(n,x),y) -> c_1(app#(x,y))
        Weak TRS Rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
        Signature:
          {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        Obligation:
          Innermost
          basic terms: {app#,reverse#,shuffle#}/{add,nil}
  *** 1.1.1.1.1.1 Progress [(?,O(n^3))]  ***
      Considered Problem:
        Strict DP Rules:
          app#(add(n,x),y) -> c_1(app#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        Weak TRS Rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
        Signature:
          {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        Obligation:
          Innermost
          basic terms: {app#,reverse#,shuffle#}/{add,nil}
      Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
      Proof:
        We decompose the input problem according to the dependency graph into the upper component
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        and a lower component
          app#(add(n,x),y) -> c_1(app#(x,y))
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        Further, following extension rules are added to the lower component.
          shuffle#(add(n,x)) -> reverse#(x)
          shuffle#(add(n,x)) -> shuffle#(reverse(x))
    *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
        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, greedy = NoGreedy}}
        Proof:
          We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
            1: shuffle#(add(n,x)) ->     
                 c_5(shuffle#(reverse(x))
                    ,reverse#(x))        
            
          The strictly oriented rules are moved into the weak component.
      *** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
          Considered Problem:
            Strict DP Rules:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
          Proof:
            We apply a matrix interpretation of kind constructor based matrix interpretation:
            The following argument positions are considered usable:
              uargs(c_5) = {1}
            
            Following symbols are considered usable:
              {app,reverse,app#,reverse#,shuffle#}
            TcT has computed the following interpretation:
                   p(add) = [1] x2 + [3]         
                   p(app) = [1] x1 + [1] x2 + [0]
                   p(nil) = [0]                  
               p(reverse) = [1] x1 + [2]         
               p(shuffle) = [1] x1 + [0]         
                  p(app#) = [1] x1 + [1] x2 + [1]
              p(reverse#) = [3]                  
              p(shuffle#) = [4] x1 + [0]         
                   p(c_1) = [1]                  
                   p(c_2) = [2]                  
                   p(c_3) = [2] x1 + [2] x2 + [4]
                   p(c_4) = [1]                  
                   p(c_5) = [1] x1 + [1] x2 + [0]
                   p(c_6) = [2]                  
            
            Following rules are strictly oriented:
            shuffle#(add(n,x)) = [4] x + [12]            
                               > [4] x + [11]            
                               = c_5(shuffle#(reverse(x))
                                    ,reverse#(x))        
            
            
            Following rules are (at-least) weakly oriented:
              app(add(n,x),y) =  [1] x + [1] y + [3]         
                              >= [1] x + [1] y + [3]         
                              =  add(n,app(x,y))             
            
                 app(nil(),y) =  [1] y + [0]                 
                              >= [1] y + [0]                 
                              =  y                           
            
            reverse(add(n,x)) =  [1] x + [5]                 
                              >= [1] x + [5]                 
                              =  app(reverse(x),add(n,nil()))
            
               reverse(nil()) =  [2]                         
                              >= [0]                         
                              =  nil()                       
            
      *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            Assumption
          Proof:
            ()
      
      *** 1.1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            RemoveWeakSuffixes
          Proof:
            Consider the dependency graph
              1:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
                 -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
              
            The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
              1: shuffle#(add(n,x)) ->     
                   c_5(shuffle#(reverse(x))
                      ,reverse#(x))        
      *** 1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            EmptyProcessor
          Proof:
            The problem is already closed. The intended complexity is O(1).
      
    *** 1.1.1.1.1.1.2 Progress [(?,O(n^2))]  ***
        Considered Problem:
          Strict DP Rules:
            app#(add(n,x),y) -> c_1(app#(x,y))
          Strict TRS Rules:
            
          Weak DP Rules:
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
        Applied Processor:
          PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
        Proof:
          We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
            1: app#(add(n,x),y) -> c_1(app#(x  
                                           ,y))
            
          The strictly oriented rules are moved into the weak component.
      *** 1.1.1.1.1.1.2.1 Progress [(?,O(n^2))]  ***
          Considered Problem:
            Strict DP Rules:
              app#(add(n,x),y) -> c_1(app#(x,y))
            Strict TRS Rules:
              
            Weak DP Rules:
              reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
              shuffle#(add(n,x)) -> reverse#(x)
              shuffle#(add(n,x)) -> shuffle#(reverse(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
          Proof:
            We apply a polynomial interpretation of kind constructor-based(mixed(2)):
            The following argument positions are considered usable:
              uargs(c_1) = {1},
              uargs(c_3) = {1,2}
            
            Following symbols are considered usable:
              {app,reverse,app#,reverse#,shuffle#}
            TcT has computed the following interpretation:
                   p(add) = 1 + x2       
                   p(app) = x1 + x2      
                   p(nil) = 0            
               p(reverse) = x1           
               p(shuffle) = 4 + x1 + x1^2
                  p(app#) = 2*x1         
              p(reverse#) = x1^2         
              p(shuffle#) = 1 + x1^2     
                   p(c_1) = x1           
                   p(c_2) = 1            
                   p(c_3) = x1 + x2      
                   p(c_4) = 0            
                   p(c_5) = 1            
                   p(c_6) = 0            
            
            Following rules are strictly oriented:
            app#(add(n,x),y) = 2 + 2*x       
                             > 2*x           
                             = c_1(app#(x,y))
            
            
            Following rules are (at-least) weakly oriented:
            reverse#(add(n,x)) =  1 + 2*x + x^2               
                               >= 2*x + x^2                   
                               =  c_3(app#(reverse(x)         
                                          ,add(n,nil()))      
                                     ,reverse#(x))            
            
            shuffle#(add(n,x)) =  2 + 2*x + x^2               
                               >= x^2                         
                               =  reverse#(x)                 
            
            shuffle#(add(n,x)) =  2 + 2*x + x^2               
                               >= 1 + x^2                     
                               =  shuffle#(reverse(x))        
            
               app(add(n,x),y) =  1 + x + y                   
                               >= 1 + x + y                   
                               =  add(n,app(x,y))             
            
                  app(nil(),y) =  y                           
                               >= y                           
                               =  y                           
            
             reverse(add(n,x)) =  1 + x                       
                               >= 1 + x                       
                               =  app(reverse(x),add(n,nil()))
            
                reverse(nil()) =  0                           
                               >= 0                           
                               =  nil()                       
            
      *** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              app#(add(n,x),y) -> c_1(app#(x,y))
              reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
              shuffle#(add(n,x)) -> reverse#(x)
              shuffle#(add(n,x)) -> shuffle#(reverse(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            Assumption
          Proof:
            ()
      
      *** 1.1.1.1.1.1.2.2 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              app#(add(n,x),y) -> c_1(app#(x,y))
              reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
              shuffle#(add(n,x)) -> reverse#(x)
              shuffle#(add(n,x)) -> shuffle#(reverse(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            RemoveWeakSuffixes
          Proof:
            Consider the dependency graph
              1:W:app#(add(n,x),y) -> c_1(app#(x,y))
                 -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
              
              2:W:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
                 -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
                 -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
              
              3:W:shuffle#(add(n,x)) -> reverse#(x)
                 -->_1 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
              
              4:W:shuffle#(add(n,x)) -> shuffle#(reverse(x))
                 -->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):4
                 -->_1 shuffle#(add(n,x)) -> reverse#(x):3
              
            The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
              4: shuffle#(add(n,x)) ->           
                   shuffle#(reverse(x))          
              3: shuffle#(add(n,x)) ->           
                   reverse#(x)                   
              2: reverse#(add(n,x)) ->           
                   c_3(app#(reverse(x)           
                           ,add(n,nil()))        
                      ,reverse#(x))              
              1: app#(add(n,x),y) -> c_1(app#(x  
                                             ,y))
      *** 1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            EmptyProcessor
          Proof:
            The problem is already closed. The intended complexity is O(1).
      
  *** 1.1.1.1.1.2 Progress [(?,O(n^2))]  ***
      Considered Problem:
        Strict DP Rules:
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          app#(add(n,x),y) -> c_1(app#(x,y))
        Weak TRS Rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
        Signature:
          {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        Obligation:
          Innermost
          basic terms: {app#,reverse#,shuffle#}/{add,nil}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):3
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
          2:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
          3:W:app#(add(n,x),y) -> c_1(app#(x,y))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: app#(add(n,x),y) -> c_1(app#(x  
                                         ,y))
  *** 1.1.1.1.1.2.1 Progress [(?,O(n^2))]  ***
      Considered Problem:
        Strict DP Rules:
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
        Signature:
          {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        Obligation:
          Innermost
          basic terms: {app#,reverse#,shuffle#}/{add,nil}
      Applied Processor:
        SimplifyRHS
      Proof:
        Consider the dependency graph
          1:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
          2:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          reverse#(add(n,x)) -> c_3(reverse#(x))
  *** 1.1.1.1.1.2.1.1 Progress [(?,O(n^2))]  ***
      Considered Problem:
        Strict DP Rules:
          reverse#(add(n,x)) -> c_3(reverse#(x))
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
        Signature:
          {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        Obligation:
          Innermost
          basic terms: {app#,reverse#,shuffle#}/{add,nil}
      Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
      Proof:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          Strict DP Rules:
            reverse#(add(n,x)) -> c_3(reverse#(x))
          Strict TRS Rules:
            
          Weak DP Rules:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
        
        Problem (S)
          Strict DP Rules:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          Strict TRS Rules:
            
          Weak DP Rules:
            reverse#(add(n,x)) -> c_3(reverse#(x))
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
    *** 1.1.1.1.1.2.1.1.1 Progress [(?,O(n^2))]  ***
        Considered Problem:
          Strict DP Rules:
            reverse#(add(n,x)) -> c_3(reverse#(x))
          Strict TRS Rules:
            
          Weak DP Rules:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
        Applied Processor:
          PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
        Proof:
          We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
            1: reverse#(add(n,x)) ->
                 c_3(reverse#(x))   
            
          The strictly oriented rules are moved into the weak component.
      *** 1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^2))]  ***
          Considered Problem:
            Strict DP Rules:
              reverse#(add(n,x)) -> c_3(reverse#(x))
            Strict TRS Rules:
              
            Weak DP Rules:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
          Proof:
            We apply a polynomial interpretation of kind constructor-based(mixed(2)):
            The following argument positions are considered usable:
              uargs(c_3) = {1},
              uargs(c_5) = {1,2}
            
            Following symbols are considered usable:
              {app,reverse,app#,reverse#,shuffle#}
            TcT has computed the following interpretation:
                   p(add) = 1 + x2       
                   p(app) = x1 + x2      
                   p(nil) = 0            
               p(reverse) = x1           
               p(shuffle) = 4 + x1 + x1^2
                  p(app#) = 2*x1^2       
              p(reverse#) = 4 + 2*x1     
              p(shuffle#) = 5 + 4*x1^2   
                   p(c_1) = x1           
                   p(c_2) = 0            
                   p(c_3) = x1           
                   p(c_4) = 0            
                   p(c_5) = x1 + x2      
                   p(c_6) = 1            
            
            Following rules are strictly oriented:
            reverse#(add(n,x)) = 6 + 2*x         
                               > 4 + 2*x         
                               = c_3(reverse#(x))
            
            
            Following rules are (at-least) weakly oriented:
            shuffle#(add(n,x)) =  9 + 8*x + 4*x^2             
                               >= 9 + 2*x + 4*x^2             
                               =  c_5(shuffle#(reverse(x))    
                                     ,reverse#(x))            
            
               app(add(n,x),y) =  1 + x + y                   
                               >= 1 + x + y                   
                               =  add(n,app(x,y))             
            
                  app(nil(),y) =  y                           
                               >= y                           
                               =  y                           
            
             reverse(add(n,x)) =  1 + x                       
                               >= 1 + x                       
                               =  app(reverse(x),add(n,nil()))
            
                reverse(nil()) =  0                           
                               >= 0                           
                               =  nil()                       
            
      *** 1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              reverse#(add(n,x)) -> c_3(reverse#(x))
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            Assumption
          Proof:
            ()
      
      *** 1.1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              reverse#(add(n,x)) -> c_3(reverse#(x))
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            RemoveWeakSuffixes
          Proof:
            Consider the dependency graph
              1:W:reverse#(add(n,x)) -> c_3(reverse#(x))
                 -->_1 reverse#(add(n,x)) -> c_3(reverse#(x)):1
              
              2:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
                 -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
                 -->_2 reverse#(add(n,x)) -> c_3(reverse#(x)):1
              
            The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
              2: shuffle#(add(n,x)) ->     
                   c_5(shuffle#(reverse(x))
                      ,reverse#(x))        
              1: reverse#(add(n,x)) ->     
                   c_3(reverse#(x))        
      *** 1.1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            EmptyProcessor
          Proof:
            The problem is already closed. The intended complexity is O(1).
      
    *** 1.1.1.1.1.2.1.1.2 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          Strict TRS Rules:
            
          Weak DP Rules:
            reverse#(add(n,x)) -> c_3(reverse#(x))
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
               -->_2 reverse#(add(n,x)) -> c_3(reverse#(x)):2
               -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
            
            2:W:reverse#(add(n,x)) -> c_3(reverse#(x))
               -->_1 reverse#(add(n,x)) -> c_3(reverse#(x)):2
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            2: reverse#(add(n,x)) ->
                 c_3(reverse#(x))   
    *** 1.1.1.1.1.2.1.1.2.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
        Applied Processor:
          SimplifyRHS
        Proof:
          Consider the dependency graph
            1:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
               -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
            
          Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
    *** 1.1.1.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
          Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
          Obligation:
            Innermost
            basic terms: {app#,reverse#,shuffle#}/{add,nil}
        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, greedy = NoGreedy}}
        Proof:
          We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
            1: shuffle#(add(n,x)) ->      
                 c_5(shuffle#(reverse(x)))
            
          The strictly oriented rules are moved into the weak component.
      *** 1.1.1.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))]  ***
          Considered Problem:
            Strict DP Rules:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
          Proof:
            We apply a matrix interpretation of kind constructor based matrix interpretation:
            The following argument positions are considered usable:
              uargs(c_5) = {1}
            
            Following symbols are considered usable:
              {app,reverse,app#,reverse#,shuffle#}
            TcT has computed the following interpretation:
                   p(add) = [1] x1 + [1] x2 + [4]
                   p(app) = [1] x1 + [1] x2 + [0]
                   p(nil) = [0]                  
               p(reverse) = [1] x1 + [0]         
               p(shuffle) = [8] x1 + [4]         
                  p(app#) = [1] x2 + [0]         
              p(reverse#) = [1]                  
              p(shuffle#) = [4] x1 + [8]         
                   p(c_1) = [1]                  
                   p(c_2) = [1]                  
                   p(c_3) = [1]                  
                   p(c_4) = [0]                  
                   p(c_5) = [1] x1 + [12]        
                   p(c_6) = [8]                  
            
            Following rules are strictly oriented:
            shuffle#(add(n,x)) = [4] n + [4] x + [24]     
                               > [4] x + [20]             
                               = c_5(shuffle#(reverse(x)))
            
            
            Following rules are (at-least) weakly oriented:
              app(add(n,x),y) =  [1] n + [1] x + [1] y + [4] 
                              >= [1] n + [1] x + [1] y + [4] 
                              =  add(n,app(x,y))             
            
                 app(nil(),y) =  [1] y + [0]                 
                              >= [1] y + [0]                 
                              =  y                           
            
            reverse(add(n,x)) =  [1] n + [1] x + [4]         
                              >= [1] n + [1] x + [4]         
                              =  app(reverse(x),add(n,nil()))
            
               reverse(nil()) =  [0]                         
                              >= [0]                         
                              =  nil()                       
            
      *** 1.1.1.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            Assumption
          Proof:
            ()
      
      *** 1.1.1.1.1.2.1.1.2.1.1.2 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            RemoveWeakSuffixes
          Proof:
            Consider the dependency graph
              1:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
                 -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x))):1
              
            The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
              1: shuffle#(add(n,x)) ->      
                   c_5(shuffle#(reverse(x)))
      *** 1.1.1.1.1.2.1.1.2.1.1.2.1 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
            Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
            Obligation:
              Innermost
              basic terms: {app#,reverse#,shuffle#}/{add,nil}
          Applied Processor:
            EmptyProcessor
          Proof:
            The problem is already closed. The intended complexity is O(1).