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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_gcd#_A(x1,x2,x3) = 2
          false_A() = 3
          s_A(x1) = x1 + 6
          gcd#_A(x1,x2) = 2
          minus_A(x1,x2) = x1 + 4
          le_A(x1,x2) = x1 + x2 + 1
          true_A() = 1
          |0|_A() = 4
          if_minus_A(x1,x2,x3) = x2 + 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_gcd#_A(x1,x2,x3) = max{26, x2 - 27, x3 - 39}
          false_A = 58
          s_A(x1) = x1 + 51
          gcd#_A(x1,x2) = max{25, x1 - 25, x2 - 39}
          minus_A(x1,x2) = x1 + 24
          le_A(x1,x2) = max{57, x1 - 1, x2 + 1}
          true_A = 63
          |0|_A = 64
          if_minus_A(x1,x2,x3) = max{39, x2 + 24}
          if_gcd#!_A(x1,x2,x3) = max{55, x1 - 2, x2 - 53, x3 - 54}
          false!_A = 54
          s!_A(x1) = 65
          gcd#!_A(x1,x2) = max{55, x1 - 52, x2 - 54}
          minus!_A(x1,x2) = max{x1 + 26, x2 + 66}
          le!_A(x1,x2) = max{x1 + 2, x2 - 23}
          true!_A = 67
          |0|!_A = 68
          if_minus!_A(x1,x2,x3) = max{65, x1 + 6, x3 + 47}
  
    partial status:
  
      pi(if_gcd#) = []
      pi(false) = []
      pi(s) = []
      pi(gcd#) = []
      pi(minus) = []
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(if_minus) = []

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: 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:

  {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: 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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gcd#_A(x1,x2) = 2
          s_A(x1) = x1 + 5
          if_gcd#_A(x1,x2,x3) = 2
          le_A(x1,x2) = x1 + x2 + 1
          true_A() = 3
          minus_A(x1,x2) = x1 + 7
          |0|_A() = 4
          false_A() = 6
          if_minus_A(x1,x2,x3) = x2 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gcd#_A(x1,x2) = max{0, x1 - 46, x2 - 45}
          s_A(x1) = x1 + 44
          if_gcd#_A(x1,x2,x3) = max{0, x1 - 185, x2 - 67, x3 - 45}
          le_A(x1,x2) = x2 + 171
          true_A = 149
          minus_A(x1,x2) = x1 + 23
          |0|_A = 14
          false_A = 0
          if_minus_A(x1,x2,x3) = max{58, x2 + 23}
          gcd#!_A(x1,x2) = max{9, x1 - 13}
          s!_A(x1) = x1 + 10
          if_gcd#!_A(x1,x2,x3) = max{6, x1 - 141, x2 - 14}
          le!_A(x1,x2) = max{6, x1 - 44}
          true!_A = 2
          minus!_A(x1,x2) = max{x1 - 10, x2 + 36}
          |0|!_A = 3
          false!_A = 4
          if_minus!_A(x1,x2,x3) = max{35, x1 - 171, x2 - 11, x3 - 1}
  
    partial status:
  
      pi(gcd#) = []
      pi(s) = []
      pi(if_gcd#) = []
      pi(le) = [1]
      pi(true) = []
      pi(minus) = [1]
      pi(|0|) = []
      pi(false) = []
      pi(if_minus) = [2]

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: 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:
  
      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 + 5
          minus#_A(x1,x2) = x1 + 2
          le_A(x1,x2) = 4
          |0|_A() = 2
          true_A() = 3
          minus_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_minus#_A(x1,x2,x3) = max{0, x1 - 13, x2 - 28, x3 - 27}
          false_A = 0
          s_A(x1) = x1 + 12
          minus#_A(x1,x2) = max{0, x1 - 28, x2 - 27}
          le_A(x1,x2) = max{12, x1 - 26, x2 - 14}
          |0|_A = 13
          true_A = 0
          minus_A(x1,x2) = max{28, x1 - 50, x2 - 50}
          if_minus#!_A(x1,x2,x3) = max{0, x2 - 53}
          false!_A = 0
          s!_A(x1) = max{0, x1 - 41}
          minus#!_A(x1,x2) = max{0, x2 - 53}
          le!_A(x1,x2) = max{0, x1 - 54}
          |0|!_A = 2
          true!_A = 3
          minus!_A(x1,x2) = max{4, x1 - 52, x2 - 35}
  
    partial status:
  
      pi(if_minus#) = []
      pi(false) = []
      pi(s) = [1]
      pi(minus#) = [1]
      pi(le) = []
      pi(|0|) = []
      pi(true) = []
      pi(minus) = []

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

  r1, r2, r4

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          le#_A(x1,x2) = x1 + x2 + 6
          s_A(x1) = x1 + 5
          le_A(x1,x2) = x1 + x2 + 6
          |0|_A() = 3
          true_A() = 2
          false_A() = 2
          minus_A(x1,x2) = x2 + 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          le#_A(x1,x2) = 0
          s_A(x1) = max{2, x1 - 13}
          le_A(x1,x2) = max{4, x1 - 3, x2 - 3}
          |0|_A = 2
          true_A = 2
          false_A = 0
          minus_A(x1,x2) = max{9, x1 - 9, x2 - 9}
          le#!_A(x1,x2) = max{3, x2 - 3}
          s!_A(x1) = max{5, x1 + 4}
          le!_A(x1,x2) = 6
          |0|!_A = 8
          true!_A = 7
          false!_A = 1
          minus!_A(x1,x2) = max{5, x2 - 16}
  
    partial status:
  
      pi(le#) = []
      pi(s) = [1]
      pi(le) = [1]
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(minus) = []

The following rules are removed:

  p1

Then no dependency pair remains.