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


-- 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: minus#(s(x),s(y)) -> minus#(x,any(y))
p4: minus#(s(x),s(y)) -> any#(y)
p5: gcd#(s(x),s(y)) -> gcd#(minus(max(x,y),min(x,y)),s(min(x,y)))
p6: gcd#(s(x),s(y)) -> minus#(max(x,y),min(x,y))
p7: gcd#(s(x),s(y)) -> max#(x,y)
p8: gcd#(s(x),s(y)) -> min#(x,y)
p9: any#(s(x)) -> any#(x)

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: minus(x,|0|()) -> x
r8: minus(s(x),s(y)) -> s(minus(x,any(y)))
r9: gcd(s(x),s(y)) -> gcd(minus(max(x,y),min(x,y)),s(min(x,y)))
r10: any(s(x)) -> s(s(any(x)))
r11: any(x) -> x

The estimated dependency graph contains the following SCCs:

  {p5}
  {p1}
  {p2}
  {p3}
  {p9}


-- Reduction pair.

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

p1: gcd#(s(x),s(y)) -> gcd#(minus(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: minus(x,|0|()) -> x
r8: minus(s(x),s(y)) -> s(minus(x,any(y)))
r9: gcd(s(x),s(y)) -> gcd(minus(max(x,y),min(x,y)),s(min(x,y)))
r10: any(s(x)) -> s(s(any(x)))
r11: any(x) -> x

The set of usable rules consists of

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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gcd#_A(x1,x2) = max{x1 + 9, x2 - 1}
        s_A(x1) = x1 + 12
        minus_A(x1,x2) = x1
        max_A(x1,x2) = max{x1 + 7, x2 + 1}
        min_A(x1,x2) = x1 + 4
        |0|_A = 3
        any_A(x1) = max{0, x1 - 25}
  
    base order 2:
  
      precedence:
      
        max > min > gcd# = |0| > minus > s > any
      
  
    partial status:
  
      pi(gcd#) = [1]
      pi(s) = []
      pi(minus) = [1]
      pi(max) = [1, 2]
      pi(min) = [1]
      pi(|0|) = []
      pi(any) = []

The following rules are removed:

  p1

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: minus(x,|0|()) -> x
r8: minus(s(x),s(y)) -> s(minus(x,any(y)))
r9: gcd(s(x),s(y)) -> gcd(minus(max(x,y),min(x,y)),s(min(x,y)))
r10: any(s(x)) -> s(s(any(x)))
r11: any(x) -> x

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        min#_A(x1,x2) = max{x1 - 6, x2 + 14}
        s_A(x1) = x1 + 13
        min_A(x1,x2) = max{x1 - 14, x2 + 30}
        |0|_A = 13
  
    base order 2:
  
      precedence:
      
        min > min# > s = |0|
      
  
    partial status:
  
      pi(min#) = [2]
      pi(s) = [1]
      pi(min) = []
      pi(|0|) = []

The following rules are removed:

  p1

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: minus(x,|0|()) -> x
r8: minus(s(x),s(y)) -> s(minus(x,any(y)))
r9: gcd(s(x),s(y)) -> gcd(minus(max(x,y),min(x,y)),s(min(x,y)))
r10: any(s(x)) -> s(s(any(x)))
r11: any(x) -> x

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        max#_A(x1,x2) = max{x1 - 1, x2 + 6}
        s_A(x1) = x1 + 5
        min_A(x1,x2) = max{6, x1 - 13, x2 - 13}
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        max# = min > s = |0|
      
  
    partial status:
  
      pi(max#) = [2]
      pi(s) = [1]
      pi(min) = []
      pi(|0|) = []

The following rules are removed:

  p1

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,any(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: minus(x,|0|()) -> x
r8: minus(s(x),s(y)) -> s(minus(x,any(y)))
r9: gcd(s(x),s(y)) -> gcd(minus(max(x,y),min(x,y)),s(min(x,y)))
r10: any(s(x)) -> s(s(any(x)))
r11: any(x) -> x

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = max{3, x1 - 14}
        s_A(x1) = max{18, x1 + 12}
        any_A(x1) = max{5, x1 - 25}
        min_A(x1,x2) = max{1, x1 - 19, x2 - 34}
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        minus# = min > s = any = |0|
      
  
    partial status:
  
      pi(minus#) = []
      pi(s) = [1]
      pi(any) = []
      pi(min) = []
      pi(|0|) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: any#(s(x)) -> any#(x)

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: minus(x,|0|()) -> x
r8: minus(s(x),s(y)) -> s(minus(x,any(y)))
r9: gcd(s(x),s(y)) -> gcd(minus(max(x,y),min(x,y)),s(min(x,y)))
r10: any(s(x)) -> s(s(any(x)))
r11: any(x) -> x

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        any#_A(x1) = max{3, x1 - 1}
        s_A(x1) = max{8, x1 + 5}
        min_A(x1,x2) = max{2, x1 - 17, x2 - 18}
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        min > s = |0| > any#
      
  
    partial status:
  
      pi(any#) = []
      pi(s) = [1]
      pi(min) = []
      pi(|0|) = []

The following rules are removed:

  p1

Then no dependency pair remains.