YES
We show the termination of the TRS R:

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


-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: ack#(s(x),s(y)) -> ack#(s(x),y)
p2: ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y)))
p3: ack#(s(x),|0|()) -> ack#(x,s(|0|()))

and R consists of:

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

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ack#_A(x1,x2) = x1 + 3
          s_A(x1) = x1 + 4
          plus_A(x1,x2) = x1 + 2
          ack_A(x1,x2) = 1
          |0|_A() = 2
  
    base order 2:
  
      precedence:
      
        plus > ack > ack# = s > |0|
      
  
    partial status:
  
      pi(ack#) = []
      pi(s) = []
      pi(plus) = [1]
      pi(ack) = []
      pi(|0|) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

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

and R consists of:

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

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

and R consists of:

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

The set of usable rules consists of

  r5

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ack#_A(x1,x2) = x1 + 1
          s_A(x1) = x1 + 2
          plus_A(x1,x2) = 1
          ack_A(x1,x2) = x2 + 4
          |0|_A() = 1
  
    base order 2:
  
      precedence:
      
        s = plus = ack > ack# = |0|
      
  
    partial status:
  
      pi(ack#) = []
      pi(s) = []
      pi(plus) = []
      pi(ack) = [2]
      pi(|0|) = []

The following rules are removed:

  p2



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

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

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ack#_A(x1,x2) = x1 + x2 + 1
          s_A(x1) = x1 + 4
  
    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: plus#(s(s(x)),y) -> plus#(x,s(y))
p2: plus#(x,s(s(y))) -> plus#(s(x),y)

and R consists of:

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

The set of usable rules consists of

  r3, r4, r5

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = x1 + x2 + 3
          s_A(x1) = x1 + 2
          plus_A(x1,x2) = x2 + 3
          |0|_A() = 1
          ack_A(x1,x2) = x1 + x2 + 2
  
    base order 2:
  
      precedence:
      
        plus# > s = plus = ack > |0|
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(plus) = []
      pi(|0|) = []
      pi(ack) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  r3, r4, r5

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = x1 + x2 + 3
          s_A(x1) = x1 + 2
          plus_A(x1,x2) = x2 + 3
          |0|_A() = 1
          ack_A(x1,x2) = x1 + x2 + 2
  
    base order 2:
  
      precedence:
      
        plus# = ack > s > plus = |0|
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(plus) = []
      pi(|0|) = []
      pi(ack) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.