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

The estimated dependency graph contains the following SCCs:

  {p3, p5, p7}
  {p1}
  {p2}


-- 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(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: gcd(|0|(),y) -> y
r7: gcd(s(x),|0|()) -> s(x)
r8: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r9: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r10: 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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            if_gcd#_A(x1,x2,x3) = x2 + x3
            false_A() = (0,0)
            s_A(x1) = x1 + (3,3)
            gcd#_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,2)
            minus_A(x1,x2) = x1 + (0,2)
            le_A(x1,x2) = (9,6)
            true_A() = (4,7)
            |0|_A() = (5,8)
    
      precedence:
      
        gcd# > minus > if_gcd# = s > |0| > false = le = true
      
      partial status:
    
        pi(if_gcd#) = [3]
        pi(false) = []
        pi(s) = [1]
        pi(gcd#) = []
        pi(minus) = [1]
        pi(le) = []
        pi(true) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            if_gcd#_A(x1,x2,x3) = x3 + (4,1)
            false_A() = (1,1)
            s_A(x1) = ((1,0),(1,0)) x1 + (2,4)
            gcd#_A(x1,x2) = (0,0)
            minus_A(x1,x2) = x1 + (5,6)
            le_A(x1,x2) = (3,2)
            true_A() = (4,3)
            |0|_A() = (5,4)
    
      precedence:
      
        if_gcd# = s = gcd# = minus = le > false > true = |0|
      
      partial status:
    
        pi(if_gcd#) = []
        pi(false) = []
        pi(s) = []
        pi(gcd#) = []
        pi(minus) = []
        pi(le) = []
        pi(true) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1, p2, p3

We remove them from the problem.  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(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: gcd(|0|(),y) -> y
r7: gcd(s(x),|0|()) -> s(x)
r8: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r9: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r10: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            le#_A(x1,x2) = ((1,0),(0,0)) x1
            s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        le# > s
      
      partial status:
    
        pi(le#) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            le#_A(x1,x2) = (0,0)
            s_A(x1) = (1,1)
    
      precedence:
      
        le# = s
      
      partial status:
    
        pi(le#) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: minus#(s(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: minus(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: gcd(|0|(),y) -> y
r7: gcd(s(x),|0|()) -> s(x)
r8: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
r9: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
r10: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            minus#_A(x1,x2) = ((1,0),(0,0)) x1
            s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        minus# > s
      
      partial status:
    
        pi(minus#) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            minus#_A(x1,x2) = (0,0)
            s_A(x1) = (1,1)
    
      precedence:
      
        minus# = s
      
      partial status:
    
        pi(minus#) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.