YES

We show the termination of the TRS R:

  ack_in(|0|(),n) -> ack_out(s(n))
  ack_in(s(m),|0|()) -> u11(ack_in(m,s(|0|())))
  u11(ack_out(n)) -> ack_out(n)
  ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
  u21(ack_out(n),m) -> u22(ack_in(m,n))
  u22(ack_out(n)) -> ack_out(n)

-- SCC decomposition.

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

p1: ack_in#(s(m),|0|()) -> u11#(ack_in(m,s(|0|())))
p2: ack_in#(s(m),|0|()) -> ack_in#(m,s(|0|()))
p3: ack_in#(s(m),s(n)) -> u21#(ack_in(s(m),n),m)
p4: ack_in#(s(m),s(n)) -> ack_in#(s(m),n)
p5: u21#(ack_out(n),m) -> u22#(ack_in(m,n))
p6: u21#(ack_out(n),m) -> ack_in#(m,n)

and R consists of:

r1: ack_in(|0|(),n) -> ack_out(s(n))
r2: ack_in(s(m),|0|()) -> u11(ack_in(m,s(|0|())))
r3: u11(ack_out(n)) -> ack_out(n)
r4: ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
r5: u21(ack_out(n),m) -> u22(ack_in(m,n))
r6: u22(ack_out(n)) -> ack_out(n)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p6}


-- Reduction pair.

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

p1: ack_in#(s(m),|0|()) -> ack_in#(m,s(|0|()))
p2: ack_in#(s(m),s(n)) -> ack_in#(s(m),n)
p3: ack_in#(s(m),s(n)) -> u21#(ack_in(s(m),n),m)
p4: u21#(ack_out(n),m) -> ack_in#(m,n)

and R consists of:

r1: ack_in(|0|(),n) -> ack_out(s(n))
r2: ack_in(s(m),|0|()) -> u11(ack_in(m,s(|0|())))
r3: u11(ack_out(n)) -> ack_out(n)
r4: ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
r5: u21(ack_out(n),m) -> u22(ack_in(m,n))
r6: u22(ack_out(n)) -> ack_out(n)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          ack_in#_A(x1,x2) = x1 + 13
          s_A(x1) = x1
          |0|_A = 1
          u21#_A(x1,x2) = x2 + 13
          ack_in_A(x1,x2) = max{18, x1 + 14, x2 + 16}
          ack_out_A(x1) = 0
          u22_A(x1) = 15
          u11_A(x1) = 0
          u21_A(x1,x2) = 15
    
      precedence:
      
        ack_in# = u21# > s = |0| = ack_in > u21 > u22 > u11 > ack_out
      
      partial status:
    
        pi(ack_in#) = [1]
        pi(s) = [1]
        pi(|0|) = []
        pi(u21#) = [2]
        pi(ack_in) = [1]
        pi(ack_out) = []
        pi(u22) = []
        pi(u11) = []
        pi(u21) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          ack_in#_A(x1,x2) = x1 + 15
          s_A(x1) = x1 + 12
          |0|_A = 9
          u21#_A(x1,x2) = x2 + 16
          ack_in_A(x1,x2) = max{18, x1 + 14}
          ack_out_A(x1) = 24
          u22_A(x1) = 38
          u11_A(x1) = 27
          u21_A(x1,x2) = 27
    
      precedence:
      
        ack_in# = s = |0| = u21# = ack_in = ack_out = u22 = u11 = u21
      
      partial status:
    
        pi(ack_in#) = [1]
        pi(s) = [1]
        pi(|0|) = []
        pi(u21#) = [2]
        pi(ack_in) = [1]
        pi(ack_out) = []
        pi(u22) = []
        pi(u11) = []
        pi(u21) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: ack_in#(s(m),s(n)) -> ack_in#(s(m),n)
p2: ack_in#(s(m),s(n)) -> u21#(ack_in(s(m),n),m)
p3: u21#(ack_out(n),m) -> ack_in#(m,n)

and R consists of:

r1: ack_in(|0|(),n) -> ack_out(s(n))
r2: ack_in(s(m),|0|()) -> u11(ack_in(m,s(|0|())))
r3: u11(ack_out(n)) -> ack_out(n)
r4: ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
r5: u21(ack_out(n),m) -> u22(ack_in(m,n))
r6: u22(ack_out(n)) -> ack_out(n)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: ack_in#(s(m),s(n)) -> ack_in#(s(m),n)
p2: ack_in#(s(m),s(n)) -> u21#(ack_in(s(m),n),m)
p3: u21#(ack_out(n),m) -> ack_in#(m,n)

and R consists of:

r1: ack_in(|0|(),n) -> ack_out(s(n))
r2: ack_in(s(m),|0|()) -> u11(ack_in(m,s(|0|())))
r3: u11(ack_out(n)) -> ack_out(n)
r4: ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
r5: u21(ack_out(n),m) -> u22(ack_in(m,n))
r6: u22(ack_out(n)) -> ack_out(n)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          ack_in#_A(x1,x2) = x1 + 5
          s_A(x1) = x1
          u21#_A(x1,x2) = x2 + 5
          ack_in_A(x1,x2) = max{x1 + 5, x2 + 4}
          ack_out_A(x1) = max{5, x1 + 4}
          u22_A(x1) = max{3, x1}
          |0|_A = 0
          u11_A(x1) = max{3, x1}
          u21_A(x1,x2) = max{x1, x2 + 5}
    
      precedence:
      
        ack_in# = s = u21# = ack_in > u22 = |0| = u11 = u21 > ack_out
      
      partial status:
    
        pi(ack_in#) = [1]
        pi(s) = [1]
        pi(u21#) = [2]
        pi(ack_in) = [1, 2]
        pi(ack_out) = [1]
        pi(u22) = []
        pi(|0|) = []
        pi(u11) = [1]
        pi(u21) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          ack_in#_A(x1,x2) = max{1, x1 - 4}
          s_A(x1) = x1 + 10
          u21#_A(x1,x2) = x2 + 1
          ack_in_A(x1,x2) = max{x1 + 21, x2 + 19}
          ack_out_A(x1) = max{25, x1}
          u22_A(x1) = 39
          |0|_A = 3
          u11_A(x1) = max{21, x1 - 1}
          u21_A(x1,x2) = max{x1 + 13, x2 + 32}
    
      precedence:
      
        ack_in# = s = u21# = ack_in = ack_out = u22 = |0| = u11 = u21
      
      partial status:
    
        pi(ack_in#) = []
        pi(s) = [1]
        pi(u21#) = [2]
        pi(ack_in) = [1, 2]
        pi(ack_out) = [1]
        pi(u22) = []
        pi(|0|) = []
        pi(u11) = []
        pi(u21) = []
    

The next rules are strictly ordered:

  p2, p3

We remove them from the problem.

-- SCC decomposition.

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

p1: ack_in#(s(m),s(n)) -> ack_in#(s(m),n)

and R consists of:

r1: ack_in(|0|(),n) -> ack_out(s(n))
r2: ack_in(s(m),|0|()) -> u11(ack_in(m,s(|0|())))
r3: u11(ack_out(n)) -> ack_out(n)
r4: ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
r5: u21(ack_out(n),m) -> u22(ack_in(m,n))
r6: u22(ack_out(n)) -> ack_out(n)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: ack_in#(s(m),s(n)) -> ack_in#(s(m),n)

and R consists of:

r1: ack_in(|0|(),n) -> ack_out(s(n))
r2: ack_in(s(m),|0|()) -> u11(ack_in(m,s(|0|())))
r3: u11(ack_out(n)) -> ack_out(n)
r4: ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
r5: u21(ack_out(n),m) -> u22(ack_in(m,n))
r6: u22(ack_out(n)) -> ack_out(n)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          ack_in#_A(x1,x2) = max{x1 - 1, x2 + 3}
          s_A(x1) = max{2, x1}
    
      precedence:
      
        ack_in# = s
      
      partial status:
    
        pi(ack_in#) = [2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          ack_in#_A(x1,x2) = max{0, x2 - 3}
          s_A(x1) = x1 + 2
    
      precedence:
      
        ack_in# = s
      
      partial status:
    
        pi(ack_in#) = []
        pi(s) = [1]
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.