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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_gcd#_A(x1,x2,x3) = max{0, x1 - 20, x2 - 32, x3 - 33}
            false_A = 7
            s_A(x1) = max{17, x1 + 12}
            gcd#_A(x1,x2) = max{0, x1 - 32, x2 - 33}
            minus_A(x1,x2) = max{31, x1 + 9}
            le_A(x1,x2) = max{6, x1 - 11, x2 - 3}
            true_A = 2
            |0|_A = 10
            pred_A(x1) = max{10, x1 - 12}
            if_gcd#!_A(x1,x2,x3) = max{3, x1 - 15, x2 - 30, x3 - 31}
            false!_A = 19
            s!_A(x1) = max{18, x1}
            gcd#!_A(x1,x2) = max{3, x1 - 29, x2 - 31}
            minus!_A(x1,x2) = max{27, x1 + 22, x2 - 18}
            le!_A(x1,x2) = 20
            true!_A = 1
            |0|!_A = 20
            pred!_A(x1) = 23
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              if_gcd#_A(x1,x2,x3) = x2 + x3
              false_A() = 3
              s_A(x1) = x1 + 7
              gcd#_A(x1,x2) = x1 + x2 + 6
              minus_A(x1,x2) = x1 + 1
              le_A(x1,x2) = 5
              true_A() = 1
              |0|_A() = 0
              pred_A(x1) = x1
              if_gcd#!_A(x1,x2,x3) = x1 + x2 + x3
              false!_A() = 19
              s!_A(x1) = x1 + 18
              gcd#!_A(x1,x2) = x1 + x2 + 6
              minus!_A(x1,x2) = 16
              le!_A(x1,x2) = x1 + 17
              true!_A() = 21
              |0|!_A() = 20
              pred!_A(x1) = 15
        
  
    partial status:
  
      pi(if_gcd#) = []
      pi(false) = []
      pi(s) = [1]
      pi(gcd#) = []
      pi(minus) = [1, 2]
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(pred) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le#_A(x1,x2) = 0
            s_A(x1) = max{7, x1 - 7}
            le#!_A(x1,x2) = 1
            s!_A(x1) = 4
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              le#_A(x1,x2) = x1 + x2
              s_A(x1) = x1 + 5
              le#!_A(x1,x2) = x1 + x2
              s!_A(x1) = 4
        
  
    partial status:
  
      pi(le#) = []
      pi(s) = []

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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus#_A(x1,x2) = 0
            s_A(x1) = max{5, x1 - 2}
            minus#!_A(x1,x2) = max{18, x1 + 1}
            s!_A(x1) = x1 + 17
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              minus#_A(x1,x2) = x2
              s_A(x1) = x1 + 1
              minus#!_A(x1,x2) = x2
              s!_A(x1) = x1 + 4
        
  
    partial status:
  
      pi(minus#) = [1]
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.