YES

We show the termination of the TRS R:

  app(nil(),y) -> y
  app(add(n,x),y) -> add(n,app(x,y))
  reverse(nil()) -> nil()
  reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
  shuffle(nil()) -> nil()
  shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(add(n,x),y) -> app#(x,y)
p2: reverse#(add(n,x)) -> app#(reverse(x),add(n,nil()))
p3: reverse#(add(n,x)) -> reverse#(x)
p4: shuffle#(add(n,x)) -> shuffle#(reverse(x))
p5: shuffle#(add(n,x)) -> reverse#(x)

and R consists of:

r1: app(nil(),y) -> y
r2: app(add(n,x),y) -> add(n,app(x,y))
r3: reverse(nil()) -> nil()
r4: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r5: shuffle(nil()) -> nil()
r6: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))

The estimated dependency graph contains the following SCCs:

  {p4}
  {p3}
  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: shuffle#(add(n,x)) -> shuffle#(reverse(x))

and R consists of:

r1: app(nil(),y) -> y
r2: app(add(n,x),y) -> add(n,app(x,y))
r3: reverse(nil()) -> nil()
r4: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r5: shuffle(nil()) -> nil()
r6: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))

The set of usable rules consists of

  r1, r2, r3, r4

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            shuffle#_A(x1) = x1
            add_A(x1,x2) = x2 + (2,0)
            reverse_A(x1) = x1 + (1,0)
            app_A(x1,x2) = ((1,0),(0,0)) x1 + x2
            nil_A() = (0,0)
    
      precedence:
      
        shuffle# = reverse > app > add = nil
      
      partial status:
    
        pi(shuffle#) = [1]
        pi(add) = []
        pi(reverse) = []
        pi(app) = [2]
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            shuffle#_A(x1) = ((0,1),(0,0)) x1 + (1,1)
            add_A(x1,x2) = (5,3)
            reverse_A(x1) = (5,2)
            app_A(x1,x2) = (5,1)
            nil_A() = (1,1)
    
      precedence:
      
        reverse > shuffle# = add = app = nil
      
      partial status:
    
        pi(shuffle#) = []
        pi(add) = []
        pi(reverse) = []
        pi(app) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: reverse#(add(n,x)) -> reverse#(x)

and R consists of:

r1: app(nil(),y) -> y
r2: app(add(n,x),y) -> add(n,app(x,y))
r3: reverse(nil()) -> nil()
r4: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r5: shuffle(nil()) -> nil()
r6: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          reverse#_A(x1) = x1 + 2
          add_A(x1,x2) = max{x1 + 2, x2 + 2}
    
      precedence:
      
        add > reverse#
      
      partial status:
    
        pi(reverse#) = [1]
        pi(add) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          reverse#_A(x1) = x1 + 3
          add_A(x1,x2) = max{x1 - 1, x2}
    
      precedence:
      
        reverse# = add
      
      partial status:
    
        pi(reverse#) = []
        pi(add) = [2]
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(add(n,x),y) -> app#(x,y)

and R consists of:

r1: app(nil(),y) -> y
r2: app(add(n,x),y) -> add(n,app(x,y))
r3: reverse(nil()) -> nil()
r4: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r5: shuffle(nil()) -> nil()
r6: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          app#_A(x1,x2) = max{x1 + 1, x2}
          add_A(x1,x2) = max{x1 - 1, x2}
    
      precedence:
      
        app# = add
      
      partial status:
    
        pi(app#) = [1, 2]
        pi(add) = [2]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          app#_A(x1,x2) = max{x1 - 1, x2 + 1}
          add_A(x1,x2) = x2
    
      precedence:
      
        app# = add
      
      partial status:
    
        pi(app#) = []
        pi(add) = [2]
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.