YES

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

  R:
  le(|0|(),y) -> true()
  le(s(x),|0|()) -> false()
  le(s(x),s(y)) -> le(x,y)
  minus(|0|(),y) -> |0|()
  minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
  if_minus(true(),s(x),y) -> |0|()
  if_minus(false(),s(x),y) -> s(minus(x,y))
  quot(|0|(),s(y)) -> |0|()
  quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
  log(s(|0|())) -> |0|()
  log(s(s(x))) -> s(log(s(quot(x,s(s(|0|()))))))

  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: le#(s(x),s(y)) -> le#(x,y)
p2: minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
p3: minus#(s(x),y) -> le#(s(x),y)
p4: if_minus#(false(),s(x),y) -> minus#(x,y)
p5: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
p6: quot#(s(x),s(y)) -> minus#(x,y)
p7: log#(s(s(x))) -> log#(s(quot(x,s(s(|0|())))))
p8: log#(s(s(x))) -> quot#(x,s(s(|0|())))

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: quot(|0|(),s(y)) -> |0|()
r9: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r10: log(s(|0|())) -> |0|()
r11: log(s(s(x))) -> s(log(s(quot(x,s(s(|0|()))))))
r12: rand(x) -> x
r13: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p7}
  {p5}
  {p2, p4}
  {p1}


-- Reduction pair.

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

p1: log#(s(s(x))) -> log#(s(quot(x,s(s(|0|())))))

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: quot(|0|(),s(y)) -> |0|()
r9: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r10: log(s(|0|())) -> |0|()
r11: log(s(s(x))) -> s(log(s(quot(x,s(s(|0|()))))))
r12: rand(x) -> x
r13: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^3
      order: lexicographic order
      interpretations:
        log#_A(x1) = ((1,0,0),(0,1,0),(0,1,0)) x1
        s_A(x1) = ((1,0,0),(0,1,0),(0,1,0)) x1 + (0,2,1)
        quot_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,1)) x1 + ((0,0,0),(0,0,0),(1,0,0)) x2 + (0,0,1)
        |0|_A() = (0,1,1)
        le_A(x1,x2) = ((0,0,0),(0,0,0),(1,0,0)) x2 + (2,2,2)
        true_A() = (1,1,3)
        false_A() = (1,1,1)
        minus_A(x1,x2) = ((1,0,0),(0,1,0),(1,0,1)) x1 + ((0,0,0),(0,0,0),(1,0,0)) x2 + (0,0,1)
        if_minus_A(x1,x2,x3) = ((1,0,0),(0,1,0),(1,0,1)) x2 + (0,0,1)
        log_A(x1) = ((1,0,0),(0,1,0),(0,1,0)) x1 + (1,0,0)
        rand_A(x1) = x1 + (1,1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        minus > quot > le > log > if_minus > s > log# > |0| > rand > false > true
      
      argument filter:
    
        pi(log#) = []
        pi(s) = []
        pi(quot) = [1]
        pi(|0|) = []
        pi(le) = []
        pi(true) = []
        pi(false) = []
        pi(minus) = [1]
        pi(if_minus) = []
        pi(log) = []
        pi(rand) = []
    

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

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: quot(|0|(),s(y)) -> |0|()
r9: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r10: log(s(|0|())) -> |0|()
r11: log(s(s(x))) -> s(log(s(quot(x,s(s(|0|()))))))
r12: rand(x) -> x
r13: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^3
      order: lexicographic order
      interpretations:
        quot#_A(x1,x2) = x1 + ((1,0,0),(0,1,0),(0,1,1)) x2
        s_A(x1) = x1 + (0,2,1)
        minus_A(x1,x2) = x1 + ((0,0,0),(0,0,0),(1,0,0)) x2 + (0,0,3)
        le_A(x1,x2) = ((0,0,0),(1,0,0),(0,1,0)) x1 + (2,2,0)
        |0|_A() = (0,1,0)
        true_A() = (0,1,2)
        false_A() = (1,1,1)
        if_minus_A(x1,x2,x3) = ((0,0,0),(0,0,0),(1,0,0)) x1 + x2 + (0,0,1)
        quot_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1
        log_A(x1) = x1 + (1,0,2)
        rand_A(x1) = x1 + (1,1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        quot > rand > le > true > false > minus > if_minus > |0| > log > s > quot#
      
      argument filter:
    
        pi(quot#) = []
        pi(s) = []
        pi(minus) = [1]
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(if_minus) = []
        pi(quot) = []
        pi(log) = []
        pi(rand) = []
    

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: if_minus#(false(),s(x),y) -> minus#(x,y)
p2: minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: quot(|0|(),s(y)) -> |0|()
r9: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r10: log(s(|0|())) -> |0|()
r11: log(s(s(x))) -> s(log(s(quot(x,s(s(|0|()))))))
r12: rand(x) -> x
r13: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^3
      order: lexicographic order
      interpretations:
        if_minus#_A(x1,x2,x3) = ((0,0,0),(1,0,0),(1,1,0)) x1 + ((1,0,0),(0,1,0),(1,1,0)) x2
        false_A() = (0,3,1)
        s_A(x1) = x1 + (0,4,1)
        minus#_A(x1,x2) = ((1,0,0),(0,1,0),(1,1,0)) x1 + (0,3,5)
        le_A(x1,x2) = (2,2,1)
        |0|_A() = (0,1,6)
        true_A() = (1,1,1)
        minus_A(x1,x2) = ((1,0,0),(0,1,0),(1,1,0)) x1 + (0,0,6)
        if_minus_A(x1,x2,x3) = ((1,0,0),(0,1,0),(1,1,0)) x2 + (0,0,3)
        quot_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1 + (0,0,6)
        log_A(x1) = ((1,0,0),(0,1,0),(0,1,0)) x1 + (1,0,0)
        rand_A(x1) = x1 + (1,1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        log > minus > if_minus > rand > quot > if_minus# > le > s > true > |0| > false > minus#
      
      argument filter:
    
        pi(if_minus#) = []
        pi(false) = []
        pi(s) = []
        pi(minus#) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(minus) = []
        pi(if_minus) = []
        pi(quot) = []
        pi(log) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1, p2

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: le#(s(x),s(y)) -> le#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(|0|(),y) -> |0|()
r5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
r6: if_minus(true(),s(x),y) -> |0|()
r7: if_minus(false(),s(x),y) -> s(minus(x,y))
r8: quot(|0|(),s(y)) -> |0|()
r9: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r10: log(s(|0|())) -> |0|()
r11: log(s(s(x))) -> s(log(s(quot(x,s(s(|0|()))))))
r12: rand(x) -> x
r13: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^3
      order: lexicographic order
      interpretations:
        le#_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1 + ((1,0,0),(0,1,0),(0,1,0)) x2
        s_A(x1) = x1 + (0,2,1)
        le_A(x1,x2) = (0,2,0)
        |0|_A() = (0,1,1)
        true_A() = (0,1,0)
        false_A() = (0,2,0)
        minus_A(x1,x2) = ((1,0,0),(0,1,0),(1,0,1)) x1 + ((0,0,0),(0,0,0),(1,0,0)) x2 + (0,0,2)
        if_minus_A(x1,x2,x3) = x1 + ((1,0,0),(0,1,0),(1,0,0)) x2 + ((0,0,0),(0,0,0),(1,0,0)) x3 + (0,0,2)
        quot_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1 + (0,0,1)
        log_A(x1) = x1 + (1,0,1)
        rand_A(x1) = ((1,0,0),(0,0,0),(1,0,0)) x1 + (1,1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        rand > minus > le > true > false > quot > if_minus > s > log > le# > |0|
      
      argument filter:
    
        pi(le#) = []
        pi(s) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(minus) = [1]
        pi(if_minus) = []
        pi(quot) = []
        pi(log) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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