YES
We show the termination of the TRS 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))


-- SCC decomposition.

Consider the 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))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

Consider the 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))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_gcd#_A(x1,x2,x3) = max{x1 + 23, x2 - 11, x3 - 22}
        false_A = 18
        s_A(x1) = x1 + 45
        gcd#_A(x1,x2) = max{19, x1 - 10, x2 - 22}
        minus_A(x1,x2) = max{50, x1, x2 + 33}
        le_A(x1,x2) = max{12, x1, x2 - 22}
        true_A = 17
        |0|_A = 19
        pred_A(x1) = max{77, x1 - 34}
        gcd_A(x1,x2) = max{40, x1 - 51, x2 + 4}
  
    base order 2:
  
      precedence:
      
        minus > pred > s = gcd > |0| > true > if_gcd# = gcd# > le > false
      
  
    partial status:
  
      pi(if_gcd#) = []
      pi(false) = []
      pi(s) = [1]
      pi(gcd#) = []
      pi(minus) = [1]
      pi(le) = [1]
      pi(true) = []
      pi(|0|) = []
      pi(pred) = []
      pi(gcd) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

Consider the 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))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the 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))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gcd#_A(x1,x2) = max{31, x1 + 13, x2 - 10}
        s_A(x1) = max{29, x1 + 23}
        if_gcd#_A(x1,x2,x3) = max{30, x1 - 2, x2 + 9, x3 - 10}
        le_A(x1,x2) = max{x1 + 1, x2 + 35}
        true_A = 34
        minus_A(x1,x2) = max{24, x1 + 9}
        |0|_A = 60
        false_A = 59
        pred_A(x1) = max{8, x1 - 2}
        gcd_A(x1,x2) = max{60, x1 - 31, x2 + 6}
  
    base order 2:
  
      precedence:
      
        gcd# = if_gcd# = |0| = gcd > true = pred > s > false > le = minus
      
  
    partial status:
  
      pi(gcd#) = [1]
      pi(s) = []
      pi(if_gcd#) = [2]
      pi(le) = [1, 2]
      pi(true) = []
      pi(minus) = [1]
      pi(|0|) = []
      pi(false) = []
      pi(pred) = []
      pi(gcd) = []

The following rules are removed:

  p2



-- SCC decomposition.

Consider the 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))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

Consider the 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))

The set of usable rules consists of

  r1, r2, r4

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le#_A(x1,x2) = max{x1 - 1, x2 + 12}
        s_A(x1) = x1 + 11
        le_A(x1,x2) = max{3, x1 - 15, x2 - 12}
        |0|_A = 2
        true_A = 1
        false_A = 1
        pred_A(x1) = max{4, x1 - 7}
  
    base order 2:
  
      precedence:
      
        le = |0| > true > le# = pred > s > false
      
  
    partial status:
  
      pi(le#) = [2]
      pi(s) = [1]
      pi(le) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(pred) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the 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))

The set of usable rules consists of

  r1, r2, r4

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = x2 + 15
        s_A(x1) = x1 + 15
        le_A(x1,x2) = max{17, x1 - 16, x2 - 5}
        |0|_A = 3
        true_A = 1
        false_A = 2
        pred_A(x1) = max{3, x1 - 14}
  
    base order 2:
  
      precedence:
      
        le = pred > s > minus# = |0| > true = false
      
  
    partial status:
  
      pi(minus#) = [2]
      pi(s) = [1]
      pi(le) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(pred) = []

The following rules are removed:

  p1

Then no dependency pair remains.