YES

We show the termination of the TRS R:

  min(x,|0|()) -> |0|()
  min(|0|(),y) -> |0|()
  min(s(x),s(y)) -> s(min(x,y))
  max(x,|0|()) -> x
  max(|0|(),y) -> y
  max(s(x),s(y)) -> s(max(x,y))
  -(x,|0|()) -> x
  -(s(x),s(y)) -> -(x,y)
  gcd(s(x),|0|()) -> s(x)
  gcd(|0|(),s(x)) -> s(x)
  gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

-- SCC decomposition.

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

p1: min#(s(x),s(y)) -> min#(x,y)
p2: max#(s(x),s(y)) -> max#(x,y)
p3: -#(s(x),s(y)) -> -#(x,y)
p4: gcd#(s(x),s(y)) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)))
p5: gcd#(s(x),s(y)) -> -#(max(x,y),min(x,y))
p6: gcd#(s(x),s(y)) -> max#(x,y)
p7: gcd#(s(x),s(y)) -> min#(x,y)

and R consists of:

r1: min(x,|0|()) -> |0|()
r2: min(|0|(),y) -> |0|()
r3: min(s(x),s(y)) -> s(min(x,y))
r4: max(x,|0|()) -> x
r5: max(|0|(),y) -> y
r6: max(s(x),s(y)) -> s(max(x,y))
r7: -(x,|0|()) -> x
r8: -(s(x),s(y)) -> -(x,y)
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(|0|(),s(x)) -> s(x)
r11: gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

The estimated dependency graph contains the following SCCs:

  {p4}
  {p1}
  {p2}
  {p3}


-- Reduction pair.

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

p1: gcd#(s(x),s(y)) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)))

and R consists of:

r1: min(x,|0|()) -> |0|()
r2: min(|0|(),y) -> |0|()
r3: min(s(x),s(y)) -> s(min(x,y))
r4: max(x,|0|()) -> x
r5: max(|0|(),y) -> y
r6: max(s(x),s(y)) -> s(max(x,y))
r7: -(x,|0|()) -> x
r8: -(s(x),s(y)) -> -(x,y)
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(|0|(),s(x)) -> s(x)
r11: gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8

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{0, x1 - 13, x2 - 19}
          s_A(x1) = max{21, x1 + 17}
          -_A(x1,x2) = max{14, x1 + 1, x2}
          max_A(x1,x2) = max{x1 + 10, x2}
          min_A(x1,x2) = max{9, x1 + 5}
          |0|_A = 4
    
      precedence:
      
        max = min > gcd# = s = - = |0|
      
      partial status:
    
        pi(gcd#) = []
        pi(s) = [1]
        pi(-) = [1, 2]
        pi(max) = [1, 2]
        pi(min) = [1]
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          gcd#_A(x1,x2) = 0
          s_A(x1) = max{11, x1 + 7}
          -_A(x1,x2) = max{x1 - 8, x2 + 24}
          max_A(x1,x2) = max{10, x1 - 8, x2 + 5}
          min_A(x1,x2) = max{10, x1 + 5}
          |0|_A = 4
    
      precedence:
      
        gcd# > s = - > max = min = |0|
      
      partial status:
    
        pi(gcd#) = []
        pi(s) = []
        pi(-) = [2]
        pi(max) = [2]
        pi(min) = [1]
        pi(|0|) = []
    

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: min#(s(x),s(y)) -> min#(x,y)

and R consists of:

r1: min(x,|0|()) -> |0|()
r2: min(|0|(),y) -> |0|()
r3: min(s(x),s(y)) -> s(min(x,y))
r4: max(x,|0|()) -> x
r5: max(|0|(),y) -> y
r6: max(s(x),s(y)) -> s(max(x,y))
r7: -(x,|0|()) -> x
r8: -(s(x),s(y)) -> -(x,y)
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(|0|(),s(x)) -> s(x)
r11: gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

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:
        
          min#_A(x1,x2) = max{2, x1 - 1, x2 + 1}
          s_A(x1) = max{1, x1}
    
      precedence:
      
        min# = s
      
      partial status:
    
        pi(min#) = [2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          min#_A(x1,x2) = 0
          s_A(x1) = max{2, x1}
    
      precedence:
      
        min# = s
      
      partial status:
    
        pi(min#) = []
        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: max#(s(x),s(y)) -> max#(x,y)

and R consists of:

r1: min(x,|0|()) -> |0|()
r2: min(|0|(),y) -> |0|()
r3: min(s(x),s(y)) -> s(min(x,y))
r4: max(x,|0|()) -> x
r5: max(|0|(),y) -> y
r6: max(s(x),s(y)) -> s(max(x,y))
r7: -(x,|0|()) -> x
r8: -(s(x),s(y)) -> -(x,y)
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(|0|(),s(x)) -> s(x)
r11: gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

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:
        
          max#_A(x1,x2) = max{2, x1 - 1, x2 + 1}
          s_A(x1) = max{1, x1}
    
      precedence:
      
        max# = s
      
      partial status:
    
        pi(max#) = [2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          max#_A(x1,x2) = 0
          s_A(x1) = max{2, x1}
    
      precedence:
      
        max# = s
      
      partial status:
    
        pi(max#) = []
        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: -#(s(x),s(y)) -> -#(x,y)

and R consists of:

r1: min(x,|0|()) -> |0|()
r2: min(|0|(),y) -> |0|()
r3: min(s(x),s(y)) -> s(min(x,y))
r4: max(x,|0|()) -> x
r5: max(|0|(),y) -> y
r6: max(s(x),s(y)) -> s(max(x,y))
r7: -(x,|0|()) -> x
r8: -(s(x),s(y)) -> -(x,y)
r9: gcd(s(x),|0|()) -> s(x)
r10: gcd(|0|(),s(x)) -> s(x)
r11: gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

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:
        
          -#_A(x1,x2) = max{2, x1 - 1, x2 + 1}
          s_A(x1) = max{1, x1}
    
      precedence:
      
        -# = s
      
      partial status:
    
        pi(-#) = [2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          -#_A(x1,x2) = 0
          s_A(x1) = max{2, x1}
    
      precedence:
      
        -# = s
      
      partial status:
    
        pi(-#) = []
        pi(s) = [1]
    

The next rules are strictly ordered:

  p1

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