YES
We show the termination of the TRS R:

  ack(|0|(),y) -> s(y)
  ack(s(x),|0|()) -> ack(x,s(|0|()))
  ack(s(x),s(y)) -> ack(x,ack(s(x),y))
  f(s(x),y) -> f(x,s(x))
  f(x,s(y)) -> f(y,x)
  f(x,y) -> ack(x,y)
  ack(s(x),y) -> f(x,x)


-- SCC decomposition.

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

p1: ack#(s(x),|0|()) -> ack#(x,s(|0|()))
p2: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p3: ack#(s(x),s(y)) -> ack#(s(x),y)
p4: f#(s(x),y) -> f#(x,s(x))
p5: f#(x,s(y)) -> f#(y,x)
p6: f#(x,y) -> ack#(x,y)
p7: ack#(s(x),y) -> f#(x,x)

and R consists of:

r1: ack(|0|(),y) -> s(y)
r2: ack(s(x),|0|()) -> ack(x,s(|0|()))
r3: ack(s(x),s(y)) -> ack(x,ack(s(x),y))
r4: f(s(x),y) -> f(x,s(x))
r5: f(x,s(y)) -> f(y,x)
r6: f(x,y) -> ack(x,y)
r7: ack(s(x),y) -> f(x,x)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

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

p1: ack#(s(x),|0|()) -> ack#(x,s(|0|()))
p2: ack#(s(x),y) -> f#(x,x)
p3: f#(x,y) -> ack#(x,y)
p4: ack#(s(x),s(y)) -> ack#(s(x),y)
p5: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p6: f#(x,s(y)) -> f#(y,x)
p7: f#(s(x),y) -> f#(x,s(x))

and R consists of:

r1: ack(|0|(),y) -> s(y)
r2: ack(s(x),|0|()) -> ack(x,s(|0|()))
r3: ack(s(x),s(y)) -> ack(x,ack(s(x),y))
r4: f(s(x),y) -> f(x,s(x))
r5: f(x,s(y)) -> f(y,x)
r6: f(x,y) -> ack(x,y)
r7: ack(s(x),y) -> f(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:
          ack#_A(x1,x2) = 1
          s_A(x1) = 2
          |0|_A() = 2
          f#_A(x1,x2) = 1
          ack_A(x1,x2) = x1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          ack#_A(x1,x2) = max{1, x1 - 15}
          s_A(x1) = max{14, x1 + 4}
          |0|_A = 3
          f#_A(x1,x2) = max{1, x1 - 15, x2 - 15}
          ack_A(x1,x2) = 5
          ack#!_A(x1,x2) = x1 + 3
          s!_A(x1) = 9
          |0|!_A = 10
          f#!_A(x1,x2) = max{x1 + 4, x2 + 3}
          ack!_A(x1,x2) = 18
  
    partial status:
  
      pi(ack#) = []
      pi(s) = []
      pi(|0|) = []
      pi(f#) = []
      pi(ack) = [1]

The following rules are removed:

  p1, p2, p3, p5, p6, p7



-- SCC decomposition.

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

p1: ack#(s(x),s(y)) -> ack#(s(x),y)

and R consists of:

r1: ack(|0|(),y) -> s(y)
r2: ack(s(x),|0|()) -> ack(x,s(|0|()))
r3: ack(s(x),s(y)) -> ack(x,ack(s(x),y))
r4: f(s(x),y) -> f(x,s(x))
r5: f(x,s(y)) -> f(y,x)
r6: f(x,y) -> ack(x,y)
r7: ack(s(x),y) -> f(x,x)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: ack#(s(x),s(y)) -> ack#(s(x),y)

and R consists of:

r1: ack(|0|(),y) -> s(y)
r2: ack(s(x),|0|()) -> ack(x,s(|0|()))
r3: ack(s(x),s(y)) -> ack(x,ack(s(x),y))
r4: f(s(x),y) -> f(x,s(x))
r5: f(x,s(y)) -> f(y,x)
r6: f(x,y) -> ack(x,y)
r7: ack(s(x),y) -> f(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:
          ack#_A(x1,x2) = x2 + 1
          s_A(x1) = x1 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          ack#_A(x1,x2) = 0
          s_A(x1) = max{1, x1 - 6}
          ack#!_A(x1,x2) = max{3, x1 - 12, x2 - 2}
          s!_A(x1) = max{2, x1 - 17}
  
    partial status:
  
      pi(ack#) = []
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.