YES

We show the termination of the relative TRS R/S:

  R:
  minus(x,|0|()) -> x
  minus(s(x),s(y)) -> minus(x,y)
  quot(|0|(),s(y)) -> |0|()
  quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
  plus(|0|(),y) -> y
  plus(s(x),y) -> s(plus(x,y))
  minus(minus(x,y),z) -> minus(x,plus(y,z))

  S:
  rand(x) -> x
  rand(x) -> rand(s(x))

-- SCC decomposition.

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

p1: minus#(s(x),s(y)) -> minus#(x,y)
p2: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
p3: quot#(s(x),s(y)) -> minus#(x,y)
p4: plus#(s(x),y) -> plus#(x,y)
p5: minus#(minus(x,y),z) -> minus#(x,plus(y,z))
p6: minus#(minus(x,y),z) -> plus#(y,z)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p2}
  {p1, p5}
  {p4}


-- Reduction pair.

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

p1: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        rand > plus > |0| > quot > minus > quot# > s
      
      argument filter:
    
        pi(quot#) = [1, 2]
        pi(s) = 1
        pi(minus) = 1
        pi(|0|) = []
        pi(quot) = 1
        pi(plus) = [1, 2]
        pi(rand) = [1]
    
    2. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        quot#_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(1,1,0,0),(0,1,1,0)) x1
        s_A(x1) = ((1,0,0,0),(0,0,0,0),(1,1,0,0),(1,0,1,0)) x1 + (1,1,4,1)
        minus_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(1,1,1,1)) x1 + (0,3,1,1)
        |0|_A() = (1,2,1,1)
        quot_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x1 + (1,3,0,0)
        plus_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(1,1,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(0,1,0,0),(1,1,0,0),(0,0,0,0)) x2 + (1,2,6,9)
        rand_A(x1) = (0,0,1,1)
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        minus > s > minus# > rand > quot > |0| > plus
      
      argument filter:
    
        pi(minus#) = [1, 2]
        pi(s) = 1
        pi(minus) = [1]
        pi(plus) = [2]
        pi(|0|) = []
        pi(quot) = []
        pi(rand) = 1
    
    2. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        minus#_A(x1,x2) = (0,0,0,0)
        s_A(x1) = (0,2,1,1)
        minus_A(x1,x2) = (1,0,1,1)
        plus_A(x1,x2) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,0)) x2 + (1,1,1,2)
        |0|_A() = (1,3,3,3)
        quot_A(x1,x2) = (1,3,2,2)
        rand_A(x1) = x1 + (1,1,1,1)
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the non-minimal 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)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the non-minimal 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)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        rand > plus > quot > s > |0| > minus > minus#
      
      argument filter:
    
        pi(minus#) = [1, 2]
        pi(s) = 1
        pi(minus) = 1
        pi(|0|) = []
        pi(quot) = [1]
        pi(plus) = [1, 2]
        pi(rand) = [1]
    
    2. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        minus#_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,0,0,0),(1,0,0,0)) x1 + ((0,0,0,0),(1,0,0,0),(1,0,0,0),(1,0,0,0)) x2
        s_A(x1) = ((1,0,0,0),(0,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + (1,0,0,1)
        minus_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,1,0,0),(1,0,0,0)) x1 + (0,1,1,1)
        |0|_A() = (1,1,1,1)
        quot_A(x1,x2) = x1 + (1,0,1,2)
        plus_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(0,0,0,0),(0,0,0,0)) x1 + x2 + (1,1,1,1)
        rand_A(x1) = (1,1,1,1)
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: rand(x) -> x
r9: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        rand > plus > minus > quot > |0| > plus# > s
      
      argument filter:
    
        pi(plus#) = 1
        pi(s) = 1
        pi(minus) = 1
        pi(|0|) = []
        pi(quot) = [1]
        pi(plus) = [1, 2]
        pi(rand) = [1]
    
    2. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        plus#_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,0,0,0),(1,1,1,0)) x1
        s_A(x1) = ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x1 + (2,1,1,1)
        minus_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(1,1,1,1)) x1 + (0,1,1,1)
        |0|_A() = (1,1,5,2)
        quot_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + (1,2,4,0)
        plus_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(0,0,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,1,0,0),(0,0,1,0),(1,1,0,1)) x2 + (1,1,3,2)
        rand_A(x1) = (0,0,1,1)
    

The next rules are strictly ordered:

  p1

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