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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          if_gcd#_A(x1,x2,x3) = x2 + x3 + (11,1)
          false_A() = (0,8)
          s_A(x1) = ((1,0),(1,1)) x1 + (0,3)
          gcd#_A(x1,x2) = x1 + x2 + (11,3)
          minus_A(x1,x2) = ((1,0),(1,1)) x1 + (0,1)
          le_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (10,2)
          true_A() = (8,8)
          |0|_A() = (7,7)
          pred_A(x1) = x1
          gcd_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (9,9)
          if_gcd_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (9,9)
          rand_A(x1) = ((1,0),(1,0)) x1 + (1,4)
  
    precedence:
    
      |0| = rand > if_gcd# = s > gcd# > false = minus = le = true = pred > gcd = if_gcd
    
    partial status:
  
      pi(if_gcd#) = []
      pi(false) = []
      pi(s) = []
      pi(gcd#) = []
      pi(minus) = []
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(pred) = []
      pi(gcd) = []
      pi(if_gcd) = []
      pi(rand) = []

The next rules are strictly ordered:

  p1

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))
p2: 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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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))
p2: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          gcd#_A(x1,x2) = x1 + (0,5)
          s_A(x1) = ((1,0),(1,1)) x1 + (0,7)
          if_gcd#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + x2 + (0,1)
          le_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (3,0)
          true_A() = (1,8)
          minus_A(x1,x2) = x1 + (0,1)
          |0|_A() = (9,0)
          false_A() = (1,0)
          pred_A(x1) = x1
          gcd_A(x1,x2) = x1 + x2 + (2,2)
          if_gcd_A(x1,x2,x3) = x2 + x3 + (2,1)
          rand_A(x1) = ((1,0),(1,0)) x1 + (1,1)
  
    precedence:
    
      minus = pred = gcd = if_gcd = rand > le > s = true > gcd# = if_gcd# = |0| > false
    
    partial status:
  
      pi(gcd#) = [1]
      pi(s) = []
      pi(if_gcd#) = [2]
      pi(le) = []
      pi(true) = []
      pi(minus) = []
      pi(|0|) = []
      pi(false) = []
      pi(pred) = []
      pi(gcd) = []
      pi(if_gcd) = []
      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: 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: 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:

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          minus#_A(x1,x2) = x2 + (1,0)
          s_A(x1) = ((1,0),(1,1)) x1 + (0,5)
          le_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (1,6)
          |0|_A() = (4,1)
          true_A() = (3,2)
          false_A() = (1,11)
          pred_A(x1) = x1
          minus_A(x1,x2) = x1 + (0,3)
          gcd_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,2)
          if_gcd_A(x1,x2,x3) = ((1,0),(1,1)) x2 + x3 + (2,1)
          rand_A(x1) = ((1,0),(1,0)) x1 + (1,6)
  
    precedence:
    
      s = true = pred = minus = gcd = if_gcd = rand > false > minus# = le = |0|
    
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(le) = [1, 2]
      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.