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(x,|0|()) -> x
  minus(s(x),s(y)) -> 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#(s(x),s(y)) -> minus#(x,y)
p3: gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
p4: gcd#(s(x),s(y)) -> le#(y,x)
p5: if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
p6: if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
p7: if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
p8: 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: minus(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: gcd(|0|(),y) -> y
r7: gcd(s(x),|0|()) -> s(x)
r8: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r9: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r10: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
r11: rand(x) -> x
r12: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p3, p5, p7}
  {p1}
  {p2}


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

The set of usable rules consists of

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

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) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
            false_A() = (1,0)
            s_A(x1) = ((1,0),(1,1)) x1
            gcd#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
            minus_A(x1,x2) = x1
            le_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,0)
            true_A() = (3,0)
            |0|_A() = (2,1)
            gcd_A(x1,x2) = x1 + x2 + (2,1)
            if_gcd_A(x1,x2,x3) = x2 + x3 + (2,1)
            rand_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        if_gcd# = false = s = gcd# = minus = le = true = |0| = gcd = if_gcd = rand
      
      partial status:
    
        pi(if_gcd#) = []
        pi(false) = []
        pi(s) = []
        pi(gcd#) = []
        pi(minus) = []
        pi(le) = []
        pi(true) = []
        pi(|0|) = []
        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) = ((1,0),(1,0)) x2 + x3 + (1,1)
            false_A() = (3,7)
            s_A(x1) = ((1,0),(1,1)) x1 + (4,3)
            gcd#_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (2,2)
            minus_A(x1,x2) = ((1,0),(1,1)) x1 + (1,1)
            le_A(x1,x2) = x2 + (2,10)
            true_A() = (3,11)
            |0|_A() = (2,4)
            gcd_A(x1,x2) = x1 + x2 + (1,8)
            if_gcd_A(x1,x2,x3) = x2 + ((1,0),(1,1)) x3 + (0,3)
            rand_A(x1) = (1,4)
    
      precedence:
      
        s = |0| > if_gcd > if_gcd# = le = rand > gcd# > gcd > false = minus = true
      
      partial status:
    
        pi(if_gcd#) = []
        pi(false) = []
        pi(s) = []
        pi(gcd#) = [2]
        pi(minus) = [1]
        pi(le) = []
        pi(true) = []
        pi(|0|) = []
        pi(gcd) = []
        pi(if_gcd) = [2]
        pi(rand) = []
    

The next rules are strictly ordered:

  p1, p3

We remove them from the problem.

-- SCC decomposition.

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

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

The set of usable rules consists of

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          le#_A(x1,x2) = max{x1 + 1, x2 + 1}
          s_A(x1) = max{2, x1}
          le_A(x1,x2) = max{x1 + 1, x2 + 1}
          |0|_A = 5
          true_A = 4
          false_A = 1
          minus_A(x1,x2) = max{x1, x2}
          gcd_A(x1,x2) = max{4, x1 + 2, x2 + 2}
          if_gcd_A(x1,x2,x3) = max{x1 + 1, x2 + 2, x3 + 2}
          rand_A(x1) = max{6, x1 + 3}
    
      precedence:
      
        le = true = minus > gcd = if_gcd > s > |0| = false > rand > le#
      
      partial status:
    
        pi(le#) = [1, 2]
        pi(s) = [1]
        pi(le) = [1, 2]
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(minus) = [1]
        pi(gcd) = []
        pi(if_gcd) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          le#_A(x1,x2) = max{x1 - 2, x2}
          s_A(x1) = max{14, x1 + 3}
          le_A(x1,x2) = max{16, x1 - 4, x2 - 4}
          |0|_A = 1
          true_A = 2
          false_A = 2
          minus_A(x1,x2) = max{28, x1 + 14}
          gcd_A(x1,x2) = 14
          if_gcd_A(x1,x2,x3) = 14
          rand_A(x1) = 1
    
      precedence:
      
        le = true = false = minus = gcd = if_gcd = rand > le# = s = |0|
      
      partial status:
    
        pi(le#) = [2]
        pi(s) = [1]
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        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#(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(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: gcd(|0|(),y) -> y
r7: gcd(s(x),|0|()) -> s(x)
r8: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r9: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r10: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
r11: rand(x) -> x
r12: rand(x) -> rand(s(x))

The set of usable rules consists of

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          minus#_A(x1,x2) = x1 + 1
          s_A(x1) = max{4, x1}
          le_A(x1,x2) = max{x1 + 2, x2 + 1}
          |0|_A = 8
          true_A = 7
          false_A = 3
          minus_A(x1,x2) = max{9, x1}
          gcd_A(x1,x2) = max{12, x1 + 3, x2 + 3}
          if_gcd_A(x1,x2,x3) = max{12, x1 + 1, x2 + 3, x3 + 3}
          rand_A(x1) = max{10, x1 + 5}
    
      precedence:
      
        minus# > |0| > true = false = minus = gcd = if_gcd > s > le > rand
      
      partial status:
    
        pi(minus#) = [1]
        pi(s) = [1]
        pi(le) = [1, 2]
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(minus) = [1]
        pi(gcd) = []
        pi(if_gcd) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          minus#_A(x1,x2) = 0
          s_A(x1) = 15
          le_A(x1,x2) = max{19, x1 - 4, x2 + 4}
          |0|_A = 2
          true_A = 3
          false_A = 5
          minus_A(x1,x2) = max{26, x1 + 11}
          gcd_A(x1,x2) = 16
          if_gcd_A(x1,x2,x3) = 16
          rand_A(x1) = 14
    
      precedence:
      
        minus = gcd = if_gcd > minus# = s = le = true = false > |0| = rand
      
      partial status:
    
        pi(minus#) = []
        pi(s) = []
        pi(le) = [2]
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        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.