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)
  pred(s(x)) -> x
  minus(x,|0|()) -> x
  minus(x,s(y)) -> pred(minus(x,y))
  gcd(|0|(),y) -> y
  gcd(s(x),|0|()) -> s(x)
  gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
  if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
  if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),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#(x,s(y)) -> pred#(minus(x,y))
p3: minus#(x,s(y)) -> minus#(x,y)
p4: gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
p5: gcd#(s(x),s(y)) -> le#(y,x)
p6: if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
p7: if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
p8: if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
p9: if_gcd#(false(),s(x),s(y)) -> minus#(y,x)

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: pred(s(x)) -> x
r5: minus(x,|0|()) -> x
r6: minus(x,s(y)) -> pred(minus(x,y))
r7: gcd(|0|(),y) -> y
r8: gcd(s(x),|0|()) -> s(x)
r9: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r10: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r11: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
r12: rand(x) -> x
r13: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p4, p6, p8}
  {p1}
  {p3}


-- Reduction pair.

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

p1: if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
p2: gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
p3: if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),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: pred(s(x)) -> x
r5: minus(x,|0|()) -> x
r6: minus(x,s(y)) -> pred(minus(x,y))
r7: gcd(|0|(),y) -> y
r8: gcd(s(x),|0|()) -> s(x)
r9: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r10: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r11: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
r12: rand(x) -> x
r13: 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

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_gcd#_A(x1,x2,x3) = x2 + x3 + (2,3)
            false_A() = (1,7)
            s_A(x1) = x1 + (0,3)
            gcd#_A(x1,x2) = x1 + x2 + (2,4)
            minus_A(x1,x2) = x1 + (0,1)
            le_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (3,2)
            true_A() = (5,5)
            |0|_A() = (3,4)
            pred_A(x1) = x1
            gcd_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (4,2)
            if_gcd_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (4,1)
            rand_A(x1) = ((1,0),(0,0)) x1 + (1,4)
    
      precedence:
      
        minus = pred = if_gcd > false = gcd# = le > if_gcd# > true = |0| = gcd > s = rand
      
      partial status:
    
        pi(if_gcd#) = [2, 3]
        pi(false) = []
        pi(s) = [1]
        pi(gcd#) = []
        pi(minus) = []
        pi(le) = [1]
        pi(true) = []
        pi(|0|) = []
        pi(pred) = []
        pi(gcd) = []
        pi(if_gcd) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            if_gcd#_A(x1,x2,x3) = x2
            false_A() = (1,1)
            s_A(x1) = x1 + (2,0)
            gcd#_A(x1,x2) = (0,0)
            minus_A(x1,x2) = (0,0)
            le_A(x1,x2) = x1 + (3,0)
            true_A() = (1,1)
            |0|_A() = (0,0)
            pred_A(x1) = (0,0)
            gcd_A(x1,x2) = (4,2)
            if_gcd_A(x1,x2,x3) = (4,2)
            rand_A(x1) = (1,1)
    
      precedence:
      
        gcd = if_gcd > true > |0| > false > rand > le > if_gcd# = s = gcd# = minus = pred
      
      partial status:
    
        pi(if_gcd#) = [2]
        pi(false) = []
        pi(s) = [1]
        pi(gcd#) = []
        pi(minus) = []
        pi(le) = [1]
        pi(true) = []
        pi(|0|) = []
        pi(pred) = []
        pi(gcd) = []
        pi(if_gcd) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p2, p3

We remove them from the problem.

-- SCC decomposition.

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

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

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: pred(s(x)) -> x
r5: minus(x,|0|()) -> x
r6: minus(x,s(y)) -> pred(minus(x,y))
r7: gcd(|0|(),y) -> y
r8: gcd(s(x),|0|()) -> s(x)
r9: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r10: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r11: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
r12: rand(x) -> x
r13: 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: pred(s(x)) -> x
r5: minus(x,|0|()) -> x
r6: minus(x,s(y)) -> pred(minus(x,y))
r7: gcd(|0|(),y) -> y
r8: gcd(s(x),|0|()) -> s(x)
r9: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r10: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r11: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
r12: rand(x) -> x
r13: 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

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) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1)
            s_A(x1) = x1 + (0,2)
            le_A(x1,x2) = (2,6)
            |0|_A() = (2,4)
            true_A() = (1,3)
            false_A() = (1,5)
            pred_A(x1) = x1
            minus_A(x1,x2) = x1 + (0,1)
            gcd_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (3,7)
            if_gcd_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (3,7)
            rand_A(x1) = ((1,0),(0,0)) x1 + (1,3)
    
      precedence:
      
        |0| = gcd = if_gcd > le# = pred = minus > le = false > true > s = rand
      
      partial status:
    
        pi(le#) = [1, 2]
        pi(s) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(pred) = []
        pi(minus) = [1]
        pi(gcd) = []
        pi(if_gcd) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            le#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (5,6)
            s_A(x1) = (4,5)
            le_A(x1,x2) = (3,4)
            |0|_A() = (2,0)
            true_A() = (1,3)
            false_A() = (3,3)
            pred_A(x1) = (0,1)
            minus_A(x1,x2) = (0,2)
            gcd_A(x1,x2) = (5,6)
            if_gcd_A(x1,x2,x3) = (5,6)
            rand_A(x1) = (1,1)
    
      precedence:
      
        le# = rand > gcd = if_gcd > pred = minus > s = le > false > |0| = true
      
      partial status:
    
        pi(le#) = []
        pi(s) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(pred) = []
        pi(minus) = []
        pi(gcd) = []
        pi(if_gcd) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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: minus#(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: pred(s(x)) -> x
r5: minus(x,|0|()) -> x
r6: minus(x,s(y)) -> pred(minus(x,y))
r7: gcd(|0|(),y) -> y
r8: gcd(s(x),|0|()) -> s(x)
r9: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r10: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r11: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
r12: rand(x) -> x
r13: 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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            minus#_A(x1,x2) = x2 + (0,3)
            s_A(x1) = x1 + (0,4)
            le_A(x1,x2) = x2 + (2,9)
            |0|_A() = (2,9)
            true_A() = (1,8)
            false_A() = (1,10)
            pred_A(x1) = x1
            minus_A(x1,x2) = x1 + (0,1)
            gcd_A(x1,x2) = x1 + x2 + (3,2)
            if_gcd_A(x1,x2,x3) = x2 + x3 + (3,1)
            rand_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        pred = minus = rand > if_gcd > gcd > s = le = |0| = false > minus# = true
      
      partial status:
    
        pi(minus#) = [2]
        pi(s) = [1]
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(pred) = []
        pi(minus) = []
        pi(gcd) = []
        pi(if_gcd) = [3]
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            minus#_A(x1,x2) = ((0,0),(1,0)) x2 + (1,3)
            s_A(x1) = x1 + (4,2)
            le_A(x1,x2) = (0,0)
            |0|_A() = (3,0)
            true_A() = (0,0)
            false_A() = (2,3)
            pred_A(x1) = (0,0)
            minus_A(x1,x2) = (0,0)
            gcd_A(x1,x2) = (1,1)
            if_gcd_A(x1,x2,x3) = (2,2)
            rand_A(x1) = (1,3)
    
      precedence:
      
        minus > rand > minus# > s = le = |0| = false = pred = if_gcd > true = gcd
      
      partial status:
    
        pi(minus#) = []
        pi(s) = [1]
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(pred) = []
        pi(minus) = []
        pi(gcd) = []
        pi(if_gcd) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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