YES

We show the termination of the relative TRS R/S:

  R:
  f(|0|()) -> true()
  f(|1|()) -> false()
  f(s(x)) -> f(x)
  if(true(),s(x),s(y)) -> s(x)
  if(false(),s(x),s(y)) -> s(y)
  g(x,c(y)) -> c(g(x,y))
  g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))

  S:
  rand(x) -> x
  rand(x) -> rand(s(x))

-- SCC decomposition.

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

p1: f#(s(x)) -> f#(x)
p2: g#(x,c(y)) -> g#(x,y)
p3: g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
p4: g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
p5: g#(x,c(y)) -> f#(x)
p6: g#(x,c(y)) -> g#(s(x),y)

and R consists of:

r1: f(|0|()) -> true()
r2: f(|1|()) -> false()
r3: f(s(x)) -> f(x)
r4: if(true(),s(x),s(y)) -> s(x)
r5: if(false(),s(x),s(y)) -> s(y)
r6: g(x,c(y)) -> c(g(x,y))
r7: g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p2, p6}
  {p1}


-- Reduction pair.

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

p1: g#(x,c(y)) -> g#(s(x),y)
p2: g#(x,c(y)) -> g#(x,y)

and R consists of:

r1: f(|0|()) -> true()
r2: f(|1|()) -> false()
r3: f(s(x)) -> f(x)
r4: if(true(),s(x),s(y)) -> s(x)
r5: if(false(),s(x),s(y)) -> s(y)
r6: g(x,c(y)) -> c(g(x,y))
r7: g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

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: lexicographic order
      interpretations:
        g#_A(x1,x2) = ((1,0),(1,1)) x2
        c_A(x1) = x1 + (2,1)
        s_A(x1) = (0,1)
        f_A(x1) = (1,2)
        |0|_A() = (0,1)
        true_A() = (0,3)
        |1|_A() = (0,1)
        false_A() = (0,1)
        if_A(x1,x2,x3) = (0,2)
        g_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (1,1)
        rand_A(x1) = ((1,0),(1,0)) x1 + (1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        rand > if > g > |1| > c > f > false > true > |0| > g# > s
      
      argument filter:
    
        pi(g#) = []
        pi(c) = []
        pi(s) = []
        pi(f) = []
        pi(|0|) = []
        pi(true) = []
        pi(|1|) = []
        pi(false) = []
        pi(if) = []
        pi(g) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: f#(s(x)) -> f#(x)

and R consists of:

r1: f(|0|()) -> true()
r2: f(|1|()) -> false()
r3: f(s(x)) -> f(x)
r4: if(true(),s(x),s(y)) -> s(x)
r5: if(false(),s(x),s(y)) -> s(y)
r6: g(x,c(y)) -> c(g(x,y))
r7: g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

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: lexicographic order
      interpretations:
        f#_A(x1) = x1
        s_A(x1) = x1 + (0,1)
        f_A(x1) = ((0,0),(1,0)) x1 + (2,2)
        |0|_A() = (0,1)
        true_A() = (1,1)
        |1|_A() = (0,1)
        false_A() = (1,3)
        if_A(x1,x2,x3) = x2 + ((1,0),(1,1)) x3 + (0,1)
        g_A(x1,x2) = x1 + (0,1)
        c_A(x1) = x1
        rand_A(x1) = ((1,0),(1,0)) x1 + (1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        rand > if > f# > g > f > c > false > |1| > s > true > |0|
      
      argument filter:
    
        pi(f#) = 1
        pi(s) = 1
        pi(f) = []
        pi(|0|) = []
        pi(true) = []
        pi(|1|) = []
        pi(false) = []
        pi(if) = [2, 3]
        pi(g) = []
        pi(c) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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