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. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (2,0)
        der_A(x1) = ((1,0),(1,0)) x1 + (1,1)
        plus_A(x1,x2) = x1 + (1,0)
        u21#_A(x1,x2,x3) = (2,0)
        din_A(x1) = x1 + (0,1)
        dout_A(x1) = ((1,1),(0,1)) x1 + (1,5)
        u41#_A(x1,x2) = (2,0)
        times_A(x1,x2) = ((0,1),(1,1)) x1 + ((0,1),(1,0)) x2 + (2,0)
        u31#_A(x1,x2,x3) = ((0,1),(0,0)) x1
        u22_A(x1,x2,x3,x4) = x4 + (2,5)
        u32_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (0,5)
        u42_A(x1,x2,x3) = x1 + (1,0)
        u21_A(x1,x2,x3) = x1 + (1,0)
        u31_A(x1,x2,x3) = x1 + ((0,1),(0,0)) x2 + (2,0)
        u41_A(x1,x2) = ((1,1),(0,0)) x1 + (0,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (0,0)
        der_A(x1) = x1 + (1,1)
        plus_A(x1,x2) = (1,0)
        u21#_A(x1,x2,x3) = (0,0)
        din_A(x1) = ((1,0),(1,0)) x1 + (1,0)
        dout_A(x1) = (5,7)
        u41#_A(x1,x2) = (0,0)
        times_A(x1,x2) = (1,1)
        u31#_A(x1,x2,x3) = (1,1)
        u22_A(x1,x2,x3,x4) = (6,7)
        u32_A(x1,x2,x3,x4) = (4,3)
        u42_A(x1,x2,x3) = (0,1)
        u21_A(x1,x2,x3) = ((0,1),(0,1)) x1
        u31_A(x1,x2,x3) = (3,2)
        u41_A(x1,x2) = (1,0)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (0,0)
        der_A(x1) = (1,1)
        plus_A(x1,x2) = (1,1)
        u21#_A(x1,x2,x3) = (0,0)
        din_A(x1) = ((0,0),(1,0)) x1 + (2,0)
        dout_A(x1) = (1,1)
        u41#_A(x1,x2) = (0,0)
        times_A(x1,x2) = (1,1)
        u31#_A(x1,x2,x3) = (1,1)
        u22_A(x1,x2,x3,x4) = (2,1)
        u32_A(x1,x2,x3,x4) = (2,3)
        u42_A(x1,x2,x3) = (0,1)
        u21_A(x1,x2,x3) = (3,2)
        u31_A(x1,x2,x3) = (3,2)
        u41_A(x1,x2) = (1,1)
    

The next rules are strictly ordered:

  p7, p8

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))) -> 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(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, p2, p3, p4, p5, p6, p7}


-- 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(plus(X,Y))) -> din#(der(X))
p4: din#(der(times(X,Y))) -> din#(der(X))
p5: din#(der(der(X))) -> u41#(din(der(X)),X)
p6: u41#(dout(DX),X) -> din#(der(DX))
p7: din#(der(der(X))) -> 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. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (2,0)
        der_A(x1) = ((0,1),(0,0)) x1 + (0,3)
        plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(0,1)) x2 + (1,4)
        u21#_A(x1,x2,x3) = ((0,1),(0,0)) x1
        din_A(x1) = x1 + (0,1)
        dout_A(x1) = (1,3)
        times_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (0,2)
        u41#_A(x1,x2) = (2,0)
        u22_A(x1,x2,x3,x4) = x1 + (2,0)
        u32_A(x1,x2,x3,x4) = ((1,1),(0,0)) x1 + (0,3)
        u42_A(x1,x2,x3) = (2,3)
        u21_A(x1,x2,x3) = (3,1)
        u31_A(x1,x2,x3) = x1 + ((0,1),(0,0)) x3 + (2,0)
        u41_A(x1,x2) = x1 + (3,0)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (1,1)
        der_A(x1) = (1,1)
        plus_A(x1,x2) = (0,1)
        u21#_A(x1,x2,x3) = (0,0)
        din_A(x1) = (0,1)
        dout_A(x1) = (1,4)
        times_A(x1,x2) = (1,1)
        u41#_A(x1,x2) = (1,1)
        u22_A(x1,x2,x3,x4) = (2,3)
        u32_A(x1,x2,x3,x4) = (1,3)
        u42_A(x1,x2,x3) = (1,1)
        u21_A(x1,x2,x3) = (1,2)
        u31_A(x1,x2,x3) = (0,1)
        u41_A(x1,x2) = (0,0)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (0,1)
        der_A(x1) = (0,1)
        plus_A(x1,x2) = (1,1)
        u21#_A(x1,x2,x3) = (1,0)
        din_A(x1) = (2,0)
        dout_A(x1) = (5,3)
        times_A(x1,x2) = (1,1)
        u41#_A(x1,x2) = (0,1)
        u22_A(x1,x2,x3,x4) = (4,2)
        u32_A(x1,x2,x3,x4) = (0,1)
        u42_A(x1,x2,x3) = (0,1)
        u21_A(x1,x2,x3) = (3,1)
        u31_A(x1,x2,x3) = (1,0)
        u41_A(x1,x2) = (1,0)
    

The next rules are strictly ordered:

  p1, p2

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))
p2: din#(der(times(X,Y))) -> din#(der(X))
p3: din#(der(der(X))) -> u41#(din(der(X)),X)
p4: u41#(dout(DX),X) -> din#(der(DX))
p5: din#(der(der(X))) -> 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, p2, p3, p4, p5}


-- Reduction pair.

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

p1: din#(der(plus(X,Y))) -> din#(der(X))
p2: din#(der(der(X))) -> din#(der(X))
p3: din#(der(der(X))) -> u41#(din(der(X)),X)
p4: u41#(dout(DX),X) -> din#(der(DX))
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. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (2,0)
        der_A(x1) = ((0,1),(1,0)) x1 + (1,2)
        plus_A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (0,1)
        u41#_A(x1,x2) = ((0,1),(0,0)) x1
        din_A(x1) = ((1,1),(0,0)) x1 + (0,1)
        dout_A(x1) = (1,3)
        times_A(x1,x2) = ((0,1),(1,1)) x1 + ((0,1),(1,0)) x2 + (0,3)
        u22_A(x1,x2,x3,x4) = x1 + (2,0)
        u32_A(x1,x2,x3,x4) = x1 + (2,0)
        u42_A(x1,x2,x3) = x1 + (2,0)
        u21_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((0,1),(0,0)) x3 + (2,0)
        u31_A(x1,x2,x3) = x1 + (2,1)
        u41_A(x1,x2) = x1 + (2,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (0,0)
        der_A(x1) = (0,1)
        plus_A(x1,x2) = ((1,1),(0,0)) x1 + (1,1)
        u41#_A(x1,x2) = (1,1)
        din_A(x1) = (1,1)
        dout_A(x1) = (4,4)
        times_A(x1,x2) = (1,1)
        u22_A(x1,x2,x3,x4) = (3,3)
        u32_A(x1,x2,x3,x4) = (5,3)
        u42_A(x1,x2,x3) = (5,3)
        u21_A(x1,x2,x3) = (2,2)
        u31_A(x1,x2,x3) = (6,2)
        u41_A(x1,x2) = (2,2)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = (1,1)
        der_A(x1) = (0,0)
        plus_A(x1,x2) = (0,1)
        u41#_A(x1,x2) = (0,0)
        din_A(x1) = (1,1)
        dout_A(x1) = (4,4)
        times_A(x1,x2) = (1,1)
        u22_A(x1,x2,x3,x4) = (3,3)
        u32_A(x1,x2,x3,x4) = (0,0)
        u42_A(x1,x2,x3) = (3,3)
        u21_A(x1,x2,x3) = (2,2)
        u31_A(x1,x2,x3) = (2,2)
        u41_A(x1,x2) = (2,2)
    

The next rules are strictly ordered:

  p3, p4

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))
p2: din#(der(der(X))) -> din#(der(X))
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: din#(der(plus(X,Y))) -> din#(der(X))
p2: din#(der(times(X,Y))) -> din#(der(X))
p3: din#(der(der(X))) -> 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. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = x1
        der_A(x1) = ((1,1),(1,0)) x1 + (1,1)
        plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
        times_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = ((1,1),(0,1)) x1
        der_A(x1) = ((1,0),(1,0)) x1 + (1,1)
        plus_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (1,1)
        times_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        din#_A(x1) = x1
        der_A(x1) = ((0,1),(0,1)) x1 + (1,1)
        plus_A(x1,x2) = x1 + ((0,0),(1,1)) x2 + (1,1)
        times_A(x1,x2) = x1 + (1,1)
    

The next rules are strictly ordered:

  p1, p2, p3

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