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, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ack#_A(x1,x2) = max{0, x1 - 5}
        s_A(x1) = x1 + 11
        |0|_A = 0
        f#_A(x1,x2) = max{x1 + 5, x2 + 4}
        ack_A(x1,x2) = 8
  
    base order 2:
  
      precedence:
      
        s > |0| = f# > ack > ack#
      
  
    partial status:
  
      pi(ack#) = []
      pi(s) = []
      pi(|0|) = []
      pi(f#) = [2]
      pi(ack) = []

The following rules are removed:

  p3



-- 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),y) -> f#(x,x)
p3: ack#(s(x),s(y)) -> ack#(s(x),y)
p4: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p5: f#(x,s(y)) -> f#(y,x)
p6: 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 estimated dependency graph contains the following SCCs:

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


-- 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),s(y)) -> ack#(x,ack(s(x),y))
p3: 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, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ack#_A(x1,x2) = max{1, x1 - 9}
        s_A(x1) = max{29, x1 + 11}
        |0|_A = 16
        ack_A(x1,x2) = max{3, x1 - 7, x2 - 12}
  
    base order 2:
  
      precedence:
      
        ack > ack# = s > |0|
      
  
    partial status:
  
      pi(ack#) = []
      pi(s) = []
      pi(|0|) = []
      pi(ack) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p2: 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, p2}


-- Reduction pair.

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

p1: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p2: 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, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ack#_A(x1,x2) = x1 + 8
        s_A(x1) = x1 + 12
        ack_A(x1,x2) = max{9, x1 - 7, x2 - 8}
  
    base order 2:
  
      precedence:
      
        s = ack > ack#
      
  
    partial status:
  
      pi(ack#) = [1]
      pi(s) = []
      pi(ack) = []

The following rules are removed:

  p1



-- 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, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ack#_A(x1,x2) = max{1, x2 - 12}
        s_A(x1) = x1 + 25
  
    base order 2:
  
      precedence:
      
        ack# = s
      
  
    partial status:
  
      pi(ack#) = []
      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: f#(x,s(y)) -> f#(y,x)
p2: 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, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1,x2) = max{x1 + 10, x2 + 3}
        s_A(x1) = x1 + 8
  
    base order 2:
  
      precedence:
      
        f# = s
      
  
    partial status:
  
      pi(f#) = [1]
      pi(s) = [1]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: f#(x,s(y)) -> f#(y,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}


-- Reduction pair.

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

p1: f#(x,s(y)) -> f#(y,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, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1,x2) = max{x1 + 2, x2 - 1}
        s_A(x1) = x1 + 4
  
    base order 2:
  
      precedence:
      
        f# > s
      
  
    partial status:
  
      pi(f#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.