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)
  minus(|0|(),y) -> |0|()
  minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
  if_minus(true(),s(x),y) -> |0|()
  if_minus(false(),s(x),y) -> s(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#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
p3: minus#(s(x),y) -> le#(s(x),y)
p4: if_minus#(false(),s(x),y) -> minus#(x,y)
p5: gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
p6: gcd#(s(x),s(y)) -> le#(y,x)
p7: if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
p8: if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
p9: if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
p10: 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(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: gcd(|0|(),y) -> y
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r11: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r12: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))

The estimated dependency graph contains the following SCCs:

  {p5, p7, p9}
  {p2, p4}
  {p1}


-- 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: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: gcd(|0|(),y) -> y
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r11: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r12: 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, >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!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              if_gcd#_A(x1,x2,x3) = x2 + x3
              false_A() = 1
              s_A(x1) = x1 + 7
              gcd#_A(x1,x2) = x1 + x2
              minus_A(x1,x2) = x1 + 5
              le_A(x1,x2) = x2 + 1
              true_A() = 1
              |0|_A() = 12
              if_minus_A(x1,x2,x3) = x2 + 5
              if_gcd#!_A(x1,x2,x3) = x2 + x3
              false!_A() = 26
              s!_A(x1) = 25
              gcd#!_A(x1,x2) = x1 + x2 + 1
              minus!_A(x1,x2) = x1 + x2 + 17
              le!_A(x1,x2) = x1 + x2 + 16
              true!_A() = 27
              |0|!_A() = 27
              if_minus!_A(x1,x2,x3) = 18
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_gcd#_A(x1,x2,x3) = max{28, x1 - 45, x2 + 1, x3}
            false_A = 46
            s_A(x1) = x1 + 21
            gcd#_A(x1,x2) = max{16, x1 + 10, x2}
            minus_A(x1,x2) = x1 + 11
            le_A(x1,x2) = max{x1 + 24, x2 + 60}
            true_A = 60
            |0|_A = 0
            if_minus_A(x1,x2,x3) = x2 + 11
            if_gcd#!_A(x1,x2,x3) = max{52, x2 + 39, x3 + 33}
            false!_A = 7
            s!_A(x1) = 2
            gcd#!_A(x1,x2) = max{x1 + 40, x2 + 34}
            minus!_A(x1,x2) = max{22, x1, x2 + 9}
            le!_A(x1,x2) = max{x1 - 7, x2 + 8}
            true!_A = 5
            |0|!_A = 5
            if_minus!_A(x1,x2,x3) = max{23, x1 + 22, x3 + 21}
        
  
    partial status:
  
      pi(if_gcd#) = []
      pi(false) = []
      pi(s) = []
      pi(gcd#) = []
      pi(minus) = [2]
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(if_minus) = [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: if_minus#(false(),s(x),y) -> minus#(x,y)
p2: minus#(s(x),y) -> if_minus#(le(s(x),y),s(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(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: gcd(|0|(),y) -> y
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r11: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r12: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))

The set of usable rules consists of

  r1, r2, r3, r4

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!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              if_minus#_A(x1,x2,x3) = x2
              false_A() = 1
              s_A(x1) = x1 + 2
              minus#_A(x1,x2) = x1
              le_A(x1,x2) = x1 + 1
              |0|_A() = 4
              true_A() = 1
              minus_A(x1,x2) = x1 + x2 + 2
              if_minus#!_A(x1,x2,x3) = 4
              false!_A() = 3
              s!_A(x1) = x1 + 2
              minus#!_A(x1,x2) = 4
              le!_A(x1,x2) = 6
              |0|!_A() = 0
              true!_A() = 1
              minus!_A(x1,x2) = 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_minus#_A(x1,x2,x3) = max{0, x1 - 16, x2 - 36, x3 - 10}
            false_A = 0
            s_A(x1) = x1 + 34
            minus#_A(x1,x2) = max{0, x1 - 36, x2 - 10}
            le_A(x1,x2) = max{15, x1 - 35, x2 - 8}
            |0|_A = 1
            true_A = 6
            minus_A(x1,x2) = max{50, x1 - 59, x2 - 59}
            if_minus#!_A(x1,x2,x3) = max{14, x2 - 22}
            false!_A = 24
            s!_A(x1) = x1 + 4
            minus#!_A(x1,x2) = max{13, x1 - 12}
            le!_A(x1,x2) = max{23, x1 - 2, x2 - 2}
            |0|!_A = 1
            true!_A = 24
            minus!_A(x1,x2) = max{7, x1 + 5, x2 - 63}
        
  
    partial status:
  
      pi(if_minus#) = []
      pi(false) = []
      pi(s) = [1]
      pi(minus#) = []
      pi(le) = [1]
      pi(|0|) = []
      pi(true) = []
      pi(minus) = [1]

The following rules are removed:

  p1, p2

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: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: gcd(|0|(),y) -> y
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r11: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r12: 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!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              le#_A(x1,x2) = x1
              s_A(x1) = x1 + 5
              le#!_A(x1,x2) = x1 + x2
              s!_A(x1) = 7
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le#_A(x1,x2) = 0
            s_A(x1) = max{11, x1 - 21}
            le#!_A(x1,x2) = max{29, x2 - 12}
            s!_A(x1) = x1 + 28
        
  
    partial status:
  
      pi(le#) = []
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.