YES

We show the termination of the TRS R:

  minus(X,s(Y)) -> pred(minus(X,Y))
  minus(X,|0|()) -> X
  pred(s(X)) -> X
  le(s(X),s(Y)) -> le(X,Y)
  le(s(X),|0|()) -> false()
  le(|0|(),Y) -> true()
  gcd(|0|(),Y) -> |0|()
  gcd(s(X),|0|()) -> s(X)
  gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
  if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
  if(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: minus#(X,s(Y)) -> pred#(minus(X,Y))
p2: minus#(X,s(Y)) -> minus#(X,Y)
p3: le#(s(X),s(Y)) -> le#(X,Y)
p4: gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y))
p5: gcd#(s(X),s(Y)) -> le#(Y,X)
p6: if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))
p7: if#(true(),s(X),s(Y)) -> minus#(X,Y)
p8: if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X))
p9: if#(false(),s(X),s(Y)) -> minus#(Y,X)

and R consists of:

r1: minus(X,s(Y)) -> pred(minus(X,Y))
r2: minus(X,|0|()) -> X
r3: pred(s(X)) -> X
r4: le(s(X),s(Y)) -> le(X,Y)
r5: le(s(X),|0|()) -> false()
r6: le(|0|(),Y) -> true()
r7: gcd(|0|(),Y) -> |0|()
r8: gcd(s(X),|0|()) -> s(X)
r9: gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
r10: if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
r11: if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))

The estimated dependency graph contains the following SCCs:

  {p4, p6, p8}
  {p2}
  {p3}


-- Reduction pair.

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

p1: gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y))
p2: if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X))
p3: if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))

and R consists of:

r1: minus(X,s(Y)) -> pred(minus(X,Y))
r2: minus(X,|0|()) -> X
r3: pred(s(X)) -> X
r4: le(s(X),s(Y)) -> le(X,Y)
r5: le(s(X),|0|()) -> false()
r6: le(|0|(),Y) -> true()
r7: gcd(|0|(),Y) -> |0|()
r8: gcd(s(X),|0|()) -> s(X)
r9: gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
r10: if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
r11: if(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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          gcd#_A(x1,x2) = max{8, x1 - 3, x2 - 5}
          s_A(x1) = x1 + 21
          if#_A(x1,x2,x3) = max{0, x1 - 1, x2 - 4, x3 - 5}
          le_A(x1,x2) = x1 + 16
          false_A = 6
          minus_A(x1,x2) = max{x1 + 18, x2 + 18}
          true_A = 16
          pred_A(x1) = max{39, x1}
          |0|_A = 17
    
      precedence:
      
        s > gcd# = if# > le > false > minus = pred = |0| > true
      
      partial status:
    
        pi(gcd#) = []
        pi(s) = [1]
        pi(if#) = []
        pi(le) = []
        pi(false) = []
        pi(minus) = [1, 2]
        pi(true) = []
        pi(pred) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          gcd#_A(x1,x2) = 1
          s_A(x1) = x1 + 7
          if#_A(x1,x2,x3) = 1
          le_A(x1,x2) = 15
          false_A = 6
          minus_A(x1,x2) = max{x1 + 13, x2 + 4}
          true_A = 16
          pred_A(x1) = 8
          |0|_A = 5
    
      precedence:
      
        gcd# = s = if# > true > le > false > minus = pred = |0|
      
      partial status:
    
        pi(gcd#) = []
        pi(s) = [1]
        pi(if#) = []
        pi(le) = []
        pi(false) = []
        pi(minus) = [1]
        pi(true) = []
        pi(pred) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y))
p2: if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))

and R consists of:

r1: minus(X,s(Y)) -> pred(minus(X,Y))
r2: minus(X,|0|()) -> X
r3: pred(s(X)) -> X
r4: le(s(X),s(Y)) -> le(X,Y)
r5: le(s(X),|0|()) -> false()
r6: le(|0|(),Y) -> true()
r7: gcd(|0|(),Y) -> |0|()
r8: gcd(s(X),|0|()) -> s(X)
r9: gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
r10: if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
r11: if(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#(le(Y,X),s(X),s(Y))
p2: if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))

and R consists of:

r1: minus(X,s(Y)) -> pred(minus(X,Y))
r2: minus(X,|0|()) -> X
r3: pred(s(X)) -> X
r4: le(s(X),s(Y)) -> le(X,Y)
r5: le(s(X),|0|()) -> false()
r6: le(|0|(),Y) -> true()
r7: gcd(|0|(),Y) -> |0|()
r8: gcd(s(X),|0|()) -> s(X)
r9: gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
r10: if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
r11: if(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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          gcd#_A(x1,x2) = x1 + 8
          s_A(x1) = x1 + 8
          if#_A(x1,x2,x3) = max{x1 + 5, x2 + 2}
          le_A(x1,x2) = x2 + 10
          true_A = 8
          minus_A(x1,x2) = x1 + 2
          pred_A(x1) = max{1, x1}
          |0|_A = 9
          false_A = 1
    
      precedence:
      
        minus > |0| > le = true > if# > gcd# = false > s = pred
      
      partial status:
    
        pi(gcd#) = [1]
        pi(s) = [1]
        pi(if#) = [1, 2]
        pi(le) = [2]
        pi(true) = []
        pi(minus) = []
        pi(pred) = []
        pi(|0|) = []
        pi(false) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          gcd#_A(x1,x2) = x1 + 14
          s_A(x1) = max{3, x1 - 3}
          if#_A(x1,x2,x3) = max{x1 + 7, x2 + 14}
          le_A(x1,x2) = max{10, x2 + 4}
          true_A = 2
          minus_A(x1,x2) = 3
          pred_A(x1) = 3
          |0|_A = 1
          false_A = 4
    
      precedence:
      
        gcd# = if# = minus > le = true = pred = false > |0| > s
      
      partial status:
    
        pi(gcd#) = []
        pi(s) = []
        pi(if#) = [1]
        pi(le) = [2]
        pi(true) = []
        pi(minus) = []
        pi(pred) = []
        pi(|0|) = []
        pi(false) = []
    

The next rules are strictly ordered:

  p1, p2

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#(X,s(Y)) -> minus#(X,Y)

and R consists of:

r1: minus(X,s(Y)) -> pred(minus(X,Y))
r2: minus(X,|0|()) -> X
r3: pred(s(X)) -> X
r4: le(s(X),s(Y)) -> le(X,Y)
r5: le(s(X),|0|()) -> false()
r6: le(|0|(),Y) -> true()
r7: gcd(|0|(),Y) -> |0|()
r8: gcd(s(X),|0|()) -> s(X)
r9: gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
r10: if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
r11: if(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:
    
        max/plus interpretations on natural numbers:
        
          minus#_A(x1,x2) = max{2, x1, x2 + 1}
          s_A(x1) = max{1, x1}
    
      precedence:
      
        minus# = s
      
      partial status:
    
        pi(minus#) = [1, 2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          minus#_A(x1,x2) = max{x1, x2 - 1}
          s_A(x1) = x1
    
      precedence:
      
        minus# = s
      
      partial status:
    
        pi(minus#) = [1]
        pi(s) = [1]
    

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: le#(s(X),s(Y)) -> le#(X,Y)

and R consists of:

r1: minus(X,s(Y)) -> pred(minus(X,Y))
r2: minus(X,|0|()) -> X
r3: pred(s(X)) -> X
r4: le(s(X),s(Y)) -> le(X,Y)
r5: le(s(X),|0|()) -> false()
r6: le(|0|(),Y) -> true()
r7: gcd(|0|(),Y) -> |0|()
r8: gcd(s(X),|0|()) -> s(X)
r9: gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
r10: if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
r11: if(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:
    
        max/plus interpretations on natural numbers:
        
          le#_A(x1,x2) = max{2, x1 - 1, x2 + 1}
          s_A(x1) = max{1, x1}
    
      precedence:
      
        le# = s
      
      partial status:
    
        pi(le#) = [2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          le#_A(x1,x2) = 0
          s_A(x1) = max{2, x1}
    
      precedence:
      
        le# = s
      
      partial status:
    
        pi(le#) = []
        pi(s) = [1]
    

The next rules are strictly ordered:

  p1

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