/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1),O(n^3))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict TRS:
            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} / {add/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            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} / {add/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            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} / {add/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          app(y,z){y -> add(x,y)} =
            app(add(x,y),z) ->^+ add(x,app(y,z))
              = C[app(y,z) = app(y,z){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            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} / {add/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        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.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - 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 TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        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()
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak DPs:
            app#(nil(),y) -> c_2()
            reverse#(nil()) -> c_4()
            shuffle#(nil()) -> c_6()
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        UsableRules
    + Details:
        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))
          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))
** Step 1.b:5: DecomposeDG.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        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))
*** Step 1.b:5.a:1: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        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)))
*** Step 1.b:5.a:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
        - Weak TRS:
            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_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        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 + [2]         
               p(app) = [1] x1 + [1] x2 + [0]
               p(nil) = [0]                  
           p(reverse) = [1] x1 + [0]         
           p(shuffle) = [1] x1 + [4]         
              p(app#) = [1] x1 + [1] x2 + [2]
          p(reverse#) = [0]                  
          p(shuffle#) = [8] x1 + [0]         
               p(c_1) = [1]                  
               p(c_2) = [1]                  
               p(c_3) = [1]                  
               p(c_4) = [1]                  
               p(c_5) = [1] x1 + [14]        
               p(c_6) = [8]                  
        
        Following rules are strictly oriented:
        shuffle#(add(n,x)) = [8] x + [16]             
                           > [8] x + [14]             
                           = c_5(shuffle#(reverse(x)))
        
        
        Following rules are (at-least) weakly oriented:
          app(add(n,x),y) =  [1] x + [1] y + [2]         
                          >= [1] x + [1] y + [2]         
                          =  add(n,app(x,y))             
        
             app(nil(),y) =  [1] y + [0]                 
                          >= [1] y + [0]                 
                          =  y                           
        
        reverse(add(n,x)) =  [1] x + [2]                 
                          >= [1] x + [2]                 
                          =  app(reverse(x),add(n,nil()))
        
           reverse(nil()) =  [0]                         
                          >= [0]                         
                          =  nil()                       
        
*** Step 1.b:5.a:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
        - Weak TRS:
            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_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: DecomposeDG.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          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))
        and a lower component
          app#(add(n,x),y) -> c_1(app#(x,y))
        Further, following extension rules are added to the lower component.
          reverse#(add(n,x)) -> app#(reverse(x),add(n,nil()))
          reverse#(add(n,x)) -> reverse#(x)
          shuffle#(add(n,x)) -> reverse#(x)
          shuffle#(add(n,x)) -> shuffle#(reverse(x))
**** Step 1.b:5.b:1.a:1: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        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:W:shuffle#(add(n,x)) -> reverse#(x)
             -->_1 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
          3:W:shuffle#(add(n,x)) -> shuffle#(reverse(x))
             -->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):3
             -->_1 shuffle#(add(n,x)) -> reverse#(x):2
          
        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))
**** Step 1.b:5.b:1.a:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_3(reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {app,reverse,app#,reverse#,shuffle#}
        TcT has computed the following interpretation:
               p(add) = [1] x2 + [8]         
               p(app) = [1] x1 + [1] x2 + [0]
               p(nil) = [0]                  
           p(reverse) = [1] x1 + [0]         
           p(shuffle) = [0]                  
              p(app#) = [1]                  
          p(reverse#) = [2] x1 + [0]         
          p(shuffle#) = [2] x1 + [0]         
               p(c_1) = [1] x1 + [4]         
               p(c_2) = [1]                  
               p(c_3) = [1] x1 + [8]         
               p(c_4) = [2]                  
               p(c_5) = [2]                  
               p(c_6) = [1]                  
        
        Following rules are strictly oriented:
        reverse#(add(n,x)) = [2] x + [16]    
                           > [2] x + [8]     
                           = c_3(reverse#(x))
        
        
        Following rules are (at-least) weakly oriented:
        shuffle#(add(n,x)) =  [2] x + [16]                
                           >= [2] x + [0]                 
                           =  reverse#(x)                 
        
        shuffle#(add(n,x)) =  [2] x + [16]                
                           >= [2] x + [0]                 
                           =  shuffle#(reverse(x))        
        
           app(add(n,x),y) =  [1] x + [1] y + [8]         
                           >= [1] x + [1] y + [8]         
                           =  add(n,app(x,y))             
        
              app(nil(),y) =  [1] y + [0]                 
                           >= [1] y + [0]                 
                           =  y                           
        
         reverse(add(n,x)) =  [1] x + [8]                 
                           >= [1] x + [8]                 
                           =  app(reverse(x),add(n,nil()))
        
            reverse(nil()) =  [0]                         
                           >= [0]                         
                           =  nil()                       
        
**** Step 1.b:5.b:1.a:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            reverse#(add(n,x)) -> c_3(reverse#(x))
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:5.b:1.b:1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
        - Weak DPs:
            reverse#(add(n,x)) -> app#(reverse(x),add(n,nil()))
            reverse#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(add) = {2},
            uargs(app) = {1},
            uargs(app#) = {1},
            uargs(shuffle#) = {1},
            uargs(c_1) = {1}
          
          Following symbols are considered usable:
            all
          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) = [1]                  
                p(app#) = [1] x1 + [1]         
            p(reverse#) = [1] x1 + [3]         
            p(shuffle#) = [1] x1 + [0]         
                 p(c_1) = [1] x1 + [1]         
                 p(c_2) = [2]                  
                 p(c_3) = [1] x1 + [1] x2 + [0]
                 p(c_4) = [2]                  
                 p(c_5) = [1] x1 + [2] x2 + [1]
                 p(c_6) = [0]                  
          
          Following rules are strictly oriented:
          app#(add(n,x),y) = [1] n + [1] x + [5]
                           > [1] x + [2]        
                           = c_1(app#(x,y))     
          
          
          Following rules are (at-least) weakly oriented:
          reverse#(add(n,x)) =  [1] n + [1] x + [7]          
                             >= [1] x + [1]                  
                             =  app#(reverse(x),add(n,nil()))
          
          reverse#(add(n,x)) =  [1] n + [1] x + [7]          
                             >= [1] x + [3]                  
                             =  reverse#(x)                  
          
          shuffle#(add(n,x)) =  [1] n + [1] x + [4]          
                             >= [1] x + [3]                  
                             =  reverse#(x)                  
          
          shuffle#(add(n,x)) =  [1] n + [1] x + [4]          
                             >= [1] x + [0]                  
                             =  shuffle#(reverse(x))         
          
             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()                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
**** Step 1.b:5.b:1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            reverse#(add(n,x)) -> app#(reverse(x),add(n,nil()))
            reverse#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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