YES

We show the termination of R/S, where R is

  le(0,y) -> true
  le(s(x),0) -> false
  le(s(x),s(y)) -> le(x,y)
  minus(0,y) -> 0
  minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
  if_minus(true,s(x),y) -> 0
  if_minus(false,s(x),y) -> s(minus(x,y))
  mod(0,y) -> 0
  mod(s(x),0) -> 0
  mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
  if_mod(true,s(x),s(y)) -> mod(minus(x,y),s(y))
  if_mod(false,s(x),s(y)) -> s(x)

and S is:

  rand(x) -> x
  rand(x) -> rand(s(x))

Since R almost dominates S and S is non-duplicating,
it is enough to show finiteness of (P, Q).
Here P consists of the dependency pairs

  le#(s(x),s(y)) -> le#(x,y)
  minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
  minus#(s(x),y) -> le#(s(x),y)
  if_minus#(false,s(x),y) -> minus#(x,y)
  mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
  mod#(s(x),s(y)) -> le#(y,x)
  if_mod#(true,s(x),s(y)) -> mod#(minus(x,y),s(y))
  if_mod#(true,s(x),s(y)) -> minus#(x,y)

and Q consists of the rules:

  le(0,y) -> true
  le(s(x),0) -> false
  le(s(x),s(y)) -> le(x,y)
  minus(0,y) -> 0
  minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
  if_minus(true,s(x),y) -> 0
  if_minus(false,s(x),y) -> s(minus(x,y))
  mod(0,y) -> 0
  mod(s(x),0) -> 0
  mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
  if_mod(true,s(x),s(y)) -> mod(minus(x,y),s(y))
  if_mod(false,s(x),s(y)) -> s(x)
  rand(x) -> x
  rand(x) -> rand(s(x))

The weakly monotone algebra (N^2, >_lex) with

  le#_A((x1,y1),(x2,y2)) = (0, 0)
  s_A((x1,y1)) = (0, 3)
  minus#_A((x1,y1),(x2,y2)) = (1, 1)
  if_minus#_A((x1,y1),(x2,y2),(x3,y3)) = (1, 1)
  le_A((x1,y1),(x2,y2)) = (2, 1)
  false_A = (1, 2)
  mod#_A((x1,y1),(x2,y2)) = (2, 2)
  if_mod#_A((x1,y1),(x2,y2),(x3,y3)) = (2, 2)
  true_A = (1, 2)
  minus_A((x1,y1),(x2,y2)) = (3, 1)
  0_A = (1, 3)
  if_minus_A((x1,y1),(x2,y2),(x3,y3)) = (2, 2)
  mod_A((x1,y1),(x2,y2)) = (2, 0)
  if_mod_A((x1,y1),(x2,y2),(x3,y3)) = (2, 0)
  rand_A((x1,y1)) = (x1 + 1, 0)

strictly orients the following dependency pairs:

  minus#(s(x),y) -> le#(s(x),y)
  mod#(s(x),s(y)) -> le#(y,x)
  if_mod#(true,s(x),s(y)) -> minus#(x,y)

We remove them from the set of dependency pairs.

The weakly monotone algebra (N^2, >_lex) with

  le#_A((x1,y1),(x2,y2)) = (x1, y1)
  s_A((x1,y1)) = (x1, y1 + 3)
  minus#_A((x1,y1),(x2,y2)) = (x1 + x2, y1 + y2 + 1)
  if_minus#_A((x1,y1),(x2,y2),(x3,y3)) = (x2 + x3, y2 + y3)
  le_A((x1,y1),(x2,y2)) = (2, 1)
  false_A = (1, 2)
  mod#_A((x1,y1),(x2,y2)) = (x1 + x2 + 2, y1 + y2 + 2)
  if_mod#_A((x1,y1),(x2,y2),(x3,y3)) = (x1 + x2 + x3, y1 + y2 + y3)
  true_A = (2, 1)
  minus_A((x1,y1),(x2,y2)) = (x1, y1 + 1)
  0_A = (0, 0)
  if_minus_A((x1,y1),(x2,y2),(x3,y3)) = (x2, y2 + 1)
  mod_A((x1,y1),(x2,y2)) = (x1 + 1, 0)
  if_mod_A((x1,y1),(x2,y2),(x3,y3)) = (x2 + 1, 0)
  rand_A((x1,y1)) = (x1 + 1, 0)

strictly orients the following dependency pairs:

  le#(s(x),s(y)) -> le#(x,y)
  minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
  if_minus#(false,s(x),y) -> minus#(x,y)
  mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
  if_mod#(true,s(x),s(y)) -> mod#(minus(x,y),s(y))

We remove them from the set of dependency pairs.

No dependency pair remains.