YES

We show the termination of the TRS R:

  minus(X,|0|()) -> X
  minus(s(X),s(Y)) -> p(minus(X,Y))
  p(s(X)) -> X
  div(|0|(),s(Y)) -> |0|()
  div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))

-- SCC decomposition.

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

p1: minus#(s(X),s(Y)) -> p#(minus(X,Y))
p2: minus#(s(X),s(Y)) -> minus#(X,Y)
p3: div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
p4: div#(s(X),s(Y)) -> minus#(X,Y)

and R consists of:

r1: minus(X,|0|()) -> X
r2: minus(s(X),s(Y)) -> p(minus(X,Y))
r3: p(s(X)) -> X
r4: div(|0|(),s(Y)) -> |0|()
r5: div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))

The estimated dependency graph contains the following SCCs:

  {p3}
  {p2}


-- Reduction pair.

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

p1: div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))

and R consists of:

r1: minus(X,|0|()) -> X
r2: minus(s(X),s(Y)) -> p(minus(X,Y))
r3: p(s(X)) -> X
r4: div(|0|(),s(Y)) -> |0|()
r5: div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        div#_A(x1,x2) = x1 + x2
        s_A(x1) = x1 + 2
        minus_A(x1,x2) = x1 + 1
        p_A(x1) = x1
        |0|_A() = 1
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        div#_A(x1,x2) = x1 + x2
        s_A(x1) = x1 + 1
        minus_A(x1,x2) = 2
        p_A(x1) = 3
        |0|_A() = 1
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  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,Y)

and R consists of:

r1: minus(X,|0|()) -> X
r2: minus(s(X),s(Y)) -> p(minus(X,Y))
r3: p(s(X)) -> X
r4: div(|0|(),s(Y)) -> |0|()
r5: div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        minus#_A(x1,x2) = x1 + x2
        s_A(x1) = x1 + 1
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        minus#_A(x1,x2) = 0
        s_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1

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