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, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gcd#_A(x1,x2) = 3
          s_A(x1) = 4
          minus_A(x1,x2) = x1 + 1
          max_A(x1,x2) = x1 + x2 + 1
          min_A(x1,x2) = x1 + 2
          |0|_A() = 0
          any_A(x1) = x1 + 6
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gcd#_A(x1,x2) = max{0, x1 - 29, x2 - 45}
          s_A(x1) = x1 + 28
          minus_A(x1,x2) = x1 + 2
          max_A(x1,x2) = max{x1, x2 + 7}
          min_A(x1,x2) = x1 + 16
          |0|_A = 16
          any_A(x1) = max{17, x1}
          gcd#!_A(x1,x2) = max{x1 + 13, x2 - 5}
          s!_A(x1) = 14
          minus!_A(x1,x2) = max{69, x1 + 41}
          max!_A(x1,x2) = max{42, x1 - 32, x2 - 29}
          min!_A(x1,x2) = 0
          |0|!_A = 1
          any!_A(x1) = max{70, x1 - 1}
  
    partial status:
  
      pi(gcd#) = []
      pi(s) = []
      pi(minus) = []
      pi(max) = [1]
      pi(min) = [1]
      pi(|0|) = []
      pi(any) = [1]

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, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          min#_A(x1,x2) = x1 + x2 + 3
          s_A(x1) = x1 + 2
          min_A(x1,x2) = x1 + x2 + 3
          |0|_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          min#_A(x1,x2) = 0
          s_A(x1) = max{2, x1 - 38}
          min_A(x1,x2) = max{21, x1 - 4, x2 - 4}
          |0|_A = 3
          min#!_A(x1,x2) = max{2, x1 - 3, x2 - 1}
          s!_A(x1) = 1
          min!_A(x1,x2) = max{4, x1 - 4, x2 - 4}
          |0|!_A = 0
  
    partial status:
  
      pi(min#) = [1]
      pi(s) = [1]
      pi(min) = [1]
      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

  (no rules)

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:
          max#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          max#_A(x1,x2) = 0
          s_A(x1) = max{5, x1 - 19}
          max#!_A(x1,x2) = max{25, x2 + 20}
          s!_A(x1) = 1
  
    partial status:
  
      pi(max#) = [1]
      pi(s) = [1]

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

  (no rules)

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:
          minus#_A(x1,x2) = x1
          s_A(x1) = x1 + 2
          any_A(x1) = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus#_A(x1,x2) = 0
          s_A(x1) = 5
          any_A(x1) = 0
          minus#!_A(x1,x2) = 0
          s!_A(x1) = 0
          any!_A(x1) = 0
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      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: 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

  (no rules)

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:
          any#_A(x1) = 0
          s_A(x1) = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          any#_A(x1) = x1
          s_A(x1) = x1 + 10
          any#!_A(x1) = max{1, x1 - 3}
          s!_A(x1) = 0
  
    partial status:
  
      pi(any#) = []
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.