YES

We show the termination of the TRS R:

  din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
  u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
  u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
  din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
  u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
  u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
  din(der(der(X))) -> u41(din(der(X)),X)
  u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
  u42(dout(DDX),X,DX) -> dout(DDX)

-- SCC decomposition.

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

p1: din#(der(plus(X,Y))) -> u21#(din(der(X)),X,Y)
p2: din#(der(plus(X,Y))) -> din#(der(X))
p3: u21#(dout(DX),X,Y) -> u22#(din(der(Y)),X,Y,DX)
p4: u21#(dout(DX),X,Y) -> din#(der(Y))
p5: din#(der(times(X,Y))) -> u31#(din(der(X)),X,Y)
p6: din#(der(times(X,Y))) -> din#(der(X))
p7: u31#(dout(DX),X,Y) -> u32#(din(der(Y)),X,Y,DX)
p8: u31#(dout(DX),X,Y) -> din#(der(Y))
p9: din#(der(der(X))) -> u41#(din(der(X)),X)
p10: din#(der(der(X))) -> din#(der(X))
p11: u41#(dout(DX),X) -> u42#(din(der(DX)),X,DX)
p12: u41#(dout(DX),X) -> din#(der(DX))

and R consists of:

r1: din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
r2: u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
r3: u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
r4: din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
r5: u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
r6: u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
r7: din(der(der(X))) -> u41(din(der(X)),X)
r8: u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
r9: u42(dout(DDX),X,DX) -> dout(DDX)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4, p5, p6, p8, p9, p10, p12}


-- Reduction pair.

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

p1: din#(der(plus(X,Y))) -> u21#(din(der(X)),X,Y)
p2: u21#(dout(DX),X,Y) -> din#(der(Y))
p3: din#(der(der(X))) -> din#(der(X))
p4: din#(der(der(X))) -> u41#(din(der(X)),X)
p5: u41#(dout(DX),X) -> din#(der(DX))
p6: din#(der(times(X,Y))) -> din#(der(X))
p7: din#(der(times(X,Y))) -> u31#(din(der(X)),X,Y)
p8: u31#(dout(DX),X,Y) -> din#(der(Y))
p9: din#(der(plus(X,Y))) -> din#(der(X))

and R consists of:

r1: din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
r2: u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
r3: u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
r4: din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
r5: u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
r6: u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
r7: din(der(der(X))) -> u41(din(der(X)),X)
r8: u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
r9: u42(dout(DDX),X,DX) -> dout(DDX)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            din#_A(x1) = (15,15)
            der_A(x1) = ((1,0),(1,0)) x1 + (16,9)
            plus_A(x1,x2) = (2,0)
            u21#_A(x1,x2,x3) = x1 + (12,1)
            din_A(x1) = (3,10)
            dout_A(x1) = ((0,0),(1,0)) x1 + (14,16)
            u41#_A(x1,x2) = ((1,0),(1,1)) x1 + (12,1)
            times_A(x1,x2) = ((1,0),(1,0)) x1 + (2,15)
            u31#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (15,1)
            u22_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + (0,19)
            u32_A(x1,x2,x3,x4) = x1 + (0,6)
            u42_A(x1,x2,x3) = x1 + (1,0)
            u21_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (3,6)
            u31_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (3,3)
            u41_A(x1,x2) = ((1,0),(1,0)) x1 + (0,6)
    
      precedence:
      
        din > u21 > din# = der = plus = u21# = dout = u41# = times = u31# = u22 = u32 = u42 = u31 = u41
      
      partial status:
    
        pi(din#) = []
        pi(der) = []
        pi(plus) = []
        pi(u21#) = [1]
        pi(din) = []
        pi(dout) = []
        pi(u41#) = [1]
        pi(times) = []
        pi(u31#) = []
        pi(u22) = []
        pi(u32) = [1]
        pi(u42) = []
        pi(u21) = []
        pi(u31) = []
        pi(u41) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            din#_A(x1) = (4,7)
            der_A(x1) = (0,5)
            plus_A(x1,x2) = (8,0)
            u21#_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (2,6)
            din_A(x1) = (2,0)
            dout_A(x1) = (6,1)
            u41#_A(x1,x2) = ((1,0),(0,0)) x1 + (1,8)
            times_A(x1,x2) = (5,8)
            u31#_A(x1,x2,x3) = (4,7)
            u22_A(x1,x2,x3,x4) = (7,0)
            u32_A(x1,x2,x3,x4) = ((0,0),(1,0)) x1 + (9,1)
            u42_A(x1,x2,x3) = (0,0)
            u21_A(x1,x2,x3) = (9,1)
            u31_A(x1,x2,x3) = (1,0)
            u41_A(x1,x2) = (1,6)
    
      precedence:
      
        der = din > u41# > din# = u31# > plus > u21# > u41 > dout = times = u22 = u42 > u31 > u32 = u21
      
      partial status:
    
        pi(din#) = []
        pi(der) = []
        pi(plus) = []
        pi(u21#) = []
        pi(din) = []
        pi(dout) = []
        pi(u41#) = []
        pi(times) = []
        pi(u31#) = []
        pi(u22) = []
        pi(u32) = []
        pi(u42) = []
        pi(u21) = []
        pi(u31) = []
        pi(u41) = []
    

The next rules are strictly ordered:

  p1, p4, p5

We remove them from the problem.

-- SCC decomposition.

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

p1: u21#(dout(DX),X,Y) -> din#(der(Y))
p2: din#(der(der(X))) -> din#(der(X))
p3: din#(der(times(X,Y))) -> din#(der(X))
p4: din#(der(times(X,Y))) -> u31#(din(der(X)),X,Y)
p5: u31#(dout(DX),X,Y) -> din#(der(Y))
p6: din#(der(plus(X,Y))) -> din#(der(X))

and R consists of:

r1: din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
r2: u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
r3: u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
r4: din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
r5: u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
r6: u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
r7: din(der(der(X))) -> u41(din(der(X)),X)
r8: u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
r9: u42(dout(DDX),X,DX) -> dout(DDX)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: din#(der(der(X))) -> din#(der(X))
p2: din#(der(plus(X,Y))) -> din#(der(X))
p3: din#(der(times(X,Y))) -> u31#(din(der(X)),X,Y)
p4: u31#(dout(DX),X,Y) -> din#(der(Y))
p5: din#(der(times(X,Y))) -> din#(der(X))

and R consists of:

r1: din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
r2: u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
r3: u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
r4: din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
r5: u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
r6: u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
r7: din(der(der(X))) -> u41(din(der(X)),X)
r8: u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
r9: u42(dout(DDX),X,DX) -> dout(DDX)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            din#_A(x1) = ((1,0),(1,1)) x1 + (13,5)
            der_A(x1) = ((1,0),(0,0)) x1 + (4,0)
            plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (18,10)
            times_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (12,9)
            u31#_A(x1,x2,x3) = x1 + ((1,0),(1,0)) x3 + (17,2)
            din_A(x1) = ((1,0),(1,0)) x1 + (6,1)
            dout_A(x1) = (1,6)
            u22_A(x1,x2,x3,x4) = (2,5)
            u32_A(x1,x2,x3,x4) = x2 + ((1,0),(1,0)) x3 + (2,7)
            u42_A(x1,x2,x3) = (2,7)
            u21_A(x1,x2,x3) = (3,0)
            u31_A(x1,x2,x3) = x2 + ((1,0),(1,0)) x3 + (11,8)
            u41_A(x1,x2) = (9,8)
    
      precedence:
      
        din# = der = plus = times = u31# = din = dout = u22 = u32 = u42 = u21 = u31 = u41
      
      partial status:
    
        pi(din#) = []
        pi(der) = []
        pi(plus) = []
        pi(times) = []
        pi(u31#) = []
        pi(din) = []
        pi(dout) = []
        pi(u22) = []
        pi(u32) = []
        pi(u42) = []
        pi(u21) = []
        pi(u31) = []
        pi(u41) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            din#_A(x1) = x1 + (1,3)
            der_A(x1) = (4,0)
            plus_A(x1,x2) = (0,0)
            times_A(x1,x2) = (10,0)
            u31#_A(x1,x2,x3) = x1 + (6,3)
            din_A(x1) = (3,2)
            dout_A(x1) = (0,1)
            u22_A(x1,x2,x3,x4) = (0,0)
            u32_A(x1,x2,x3,x4) = (0,0)
            u42_A(x1,x2,x3) = (1,2)
            u21_A(x1,x2,x3) = (0,0)
            u31_A(x1,x2,x3) = x2
            u41_A(x1,x2) = (2,1)
    
      precedence:
      
        din > der = plus = u32 = u42 = u31 > u22 > din# = times = u31# = dout = u21 = u41
      
      partial status:
    
        pi(din#) = [1]
        pi(der) = []
        pi(plus) = []
        pi(times) = []
        pi(u31#) = [1]
        pi(din) = []
        pi(dout) = []
        pi(u22) = []
        pi(u32) = []
        pi(u42) = []
        pi(u21) = []
        pi(u31) = [2]
        pi(u41) = []
    

The next rules are strictly ordered:

  p1, p3, p4, p5

We remove them from the problem.

-- SCC decomposition.

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

p1: din#(der(plus(X,Y))) -> din#(der(X))

and R consists of:

r1: din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
r2: u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
r3: u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
r4: din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
r5: u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
r6: u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
r7: din(der(der(X))) -> u41(din(der(X)),X)
r8: u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
r9: u42(dout(DDX),X,DX) -> dout(DDX)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: din#(der(plus(X,Y))) -> din#(der(X))

and R consists of:

r1: din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
r2: u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
r3: u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
r4: din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
r5: u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
r6: u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
r7: din(der(der(X))) -> u41(din(der(X)),X)
r8: u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
r9: u42(dout(DDX),X,DX) -> dout(DDX)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            din#_A(x1) = ((0,0),(1,0)) x1 + (4,1)
            der_A(x1) = x1 + (3,7)
            plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,8)
    
      precedence:
      
        plus > din# = der
      
      partial status:
    
        pi(din#) = []
        pi(der) = []
        pi(plus) = [2]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            din#_A(x1) = (1,1)
            der_A(x1) = (0,0)
            plus_A(x1,x2) = (2,2)
    
      precedence:
      
        plus > din# = der
      
      partial status:
    
        pi(din#) = []
        pi(der) = []
        pi(plus) = []
    

The next rules are strictly ordered:

  p1

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