YES

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

  R:
  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)

  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: le#(s(x),s(y)) -> le#(x,y)
p2: minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
p3: minus#(s(x),y) -> le#(s(x),y)
p4: if_minus#(false(),s(x),y) -> minus#(x,y)
p5: mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
p6: mod#(s(x),s(y)) -> le#(y,x)
p7: if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
p8: if_mod#(true(),s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: mod(|0|(),y) -> |0|()
r9: mod(s(x),|0|()) -> |0|()
r10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r11: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r12: if_mod(false(),s(x),s(y)) -> s(x)
r13: rand(x) -> x
r14: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p5, p7}
  {p2, p4}
  {p1}


-- Reduction pair.

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

p1: if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
p2: mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: mod(|0|(),y) -> |0|()
r9: mod(s(x),|0|()) -> |0|()
r10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r11: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r12: if_mod(false(),s(x),s(y)) -> s(x)
r13: rand(x) -> x
r14: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            if_mod#_A(x1,x2,x3) = x2 + ((1,0),(1,0)) x3 + (1,1)
            true_A() = (0,1)
            s_A(x1) = x1 + (0,7)
            mod#_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (1,2)
            minus_A(x1,x2) = x1 + (0,4)
            le_A(x1,x2) = (0,3)
            |0|_A() = (0,2)
            false_A() = (0,3)
            if_minus_A(x1,x2,x3) = x1 + x2 + (0,1)
            mod_A(x1,x2) = x1 + (0,4)
            if_mod_A(x1,x2,x3) = x2 + (0,4)
            rand_A(x1) = ((1,0),(0,0)) x1 + (1,8)
    
      precedence:
      
        minus > s = |0| = if_minus = mod = if_mod > le = false > true > mod# > if_mod# > rand
      
      partial status:
    
        pi(if_mod#) = [2]
        pi(true) = []
        pi(s) = []
        pi(mod#) = []
        pi(minus) = [1]
        pi(le) = []
        pi(|0|) = []
        pi(false) = []
        pi(if_minus) = [1, 2]
        pi(mod) = []
        pi(if_mod) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            if_mod#_A(x1,x2,x3) = (4,1)
            true_A() = (6,3)
            s_A(x1) = (5,2)
            mod#_A(x1,x2) = (3,4)
            minus_A(x1,x2) = (8,4)
            le_A(x1,x2) = (2,1)
            |0|_A() = (0,0)
            false_A() = (1,1)
            if_minus_A(x1,x2,x3) = x2 + (2,1)
            mod_A(x1,x2) = (7,5)
            if_mod_A(x1,x2,x3) = (7,5)
            rand_A(x1) = (5,3)
    
      precedence:
      
        rand > minus = if_minus > |0| > le > true > if_mod# = s = mod = if_mod > mod# > false
      
      partial status:
    
        pi(if_mod#) = []
        pi(true) = []
        pi(s) = []
        pi(mod#) = []
        pi(minus) = []
        pi(le) = []
        pi(|0|) = []
        pi(false) = []
        pi(if_minus) = [2]
        pi(mod) = []
        pi(if_mod) = []
        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: if_minus#(false(),s(x),y) -> minus#(x,y)
p2: minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: mod(|0|(),y) -> |0|()
r9: mod(s(x),|0|()) -> |0|()
r10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r11: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r12: if_mod(false(),s(x),s(y)) -> s(x)
r13: rand(x) -> x
r14: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            if_minus#_A(x1,x2,x3) = ((1,0),(1,0)) x1 + x2 + ((1,0),(1,1)) x3 + (1,1)
            false_A() = (2,4)
            s_A(x1) = x1 + (0,3)
            minus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (3,5)
            le_A(x1,x2) = (2,4)
            |0|_A() = (0,0)
            true_A() = (1,3)
            minus_A(x1,x2) = x1 + (0,2)
            if_minus_A(x1,x2,x3) = ((0,0),(1,0)) x1 + x2
            mod_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,2)
            if_mod_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (3,2)
            rand_A(x1) = ((1,0),(0,0)) x1 + (1,3)
    
      precedence:
      
        mod = if_mod = rand > |0| = minus = if_minus > s > le > false > if_minus# = minus# = true
      
      partial status:
    
        pi(if_minus#) = [2]
        pi(false) = []
        pi(s) = []
        pi(minus#) = [1, 2]
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(minus) = [1]
        pi(if_minus) = []
        pi(mod) = []
        pi(if_mod) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            if_minus#_A(x1,x2,x3) = x2 + (1,2)
            false_A() = (11,1)
            s_A(x1) = (2,3)
            minus#_A(x1,x2) = (1,4)
            le_A(x1,x2) = (10,2)
            |0|_A() = (5,11)
            true_A() = (4,4)
            minus_A(x1,x2) = ((1,0),(1,0)) x1 + (7,5)
            if_minus_A(x1,x2,x3) = (6,6)
            mod_A(x1,x2) = (3,1)
            if_mod_A(x1,x2,x3) = (3,1)
            rand_A(x1) = (0,0)
    
      precedence:
      
        false = true > minus > le > mod = if_mod > if_minus > |0| > if_minus# = s = minus# = rand
      
      partial status:
    
        pi(if_minus#) = []
        pi(false) = []
        pi(s) = []
        pi(minus#) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(minus) = []
        pi(if_minus) = []
        pi(mod) = []
        pi(if_mod) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: if_minus#(false(),s(x),y) -> minus#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: mod(|0|(),y) -> |0|()
r9: mod(s(x),|0|()) -> |0|()
r10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r11: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r12: if_mod(false(),s(x),s(y)) -> s(x)
r13: rand(x) -> x
r14: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: le#(s(x),s(y)) -> le#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: mod(|0|(),y) -> |0|()
r9: mod(s(x),|0|()) -> |0|()
r10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r11: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r12: if_mod(false(),s(x),s(y)) -> s(x)
r13: rand(x) -> x
r14: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            le#_A(x1,x2) = x1 + (0,1)
            s_A(x1) = x1 + (0,6)
            le_A(x1,x2) = (3,8)
            |0|_A() = (0,5)
            true_A() = (1,7)
            false_A() = (3,4)
            minus_A(x1,x2) = x1 + (0,3)
            if_minus_A(x1,x2,x3) = ((0,0),(1,0)) x1 + x2
            mod_A(x1,x2) = x1 + (4,3)
            if_mod_A(x1,x2,x3) = x2 + (4,0)
            rand_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        le# = minus > le > true = false = mod = if_mod > if_minus > s = |0| = rand
      
      partial status:
    
        pi(le#) = [1]
        pi(s) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(minus) = [1]
        pi(if_minus) = []
        pi(mod) = []
        pi(if_mod) = [2]
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            le#_A(x1,x2) = (2,1)
            s_A(x1) = (3,2)
            le_A(x1,x2) = (5,6)
            |0|_A() = (1,0)
            true_A() = (1,0)
            false_A() = (2,3)
            minus_A(x1,x2) = ((1,0),(1,1)) x1
            if_minus_A(x1,x2,x3) = (3,3)
            mod_A(x1,x2) = (4,3)
            if_mod_A(x1,x2,x3) = x2
            rand_A(x1) = (0,0)
    
      precedence:
      
        le# = le = rand > mod > if_mod > s = if_minus > |0| = true = false = minus
      
      partial status:
    
        pi(le#) = []
        pi(s) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(minus) = [1]
        pi(if_minus) = []
        pi(mod) = []
        pi(if_mod) = [2]
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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