YES
We show the termination of the TRS R:

  plus(x,|0|()) -> x
  plus(|0|(),y) -> y
  plus(s(x),y) -> s(plus(x,y))
  times(|0|(),y) -> |0|()
  times(s(|0|()),y) -> y
  times(s(x),y) -> plus(y,times(x,y))
  div(|0|(),y) -> |0|()
  div(x,y) -> quot(x,y,y)
  quot(|0|(),s(y),z) -> |0|()
  quot(s(x),s(y),z) -> quot(x,y,z)
  quot(x,|0|(),s(z)) -> s(div(x,s(z)))
  div(div(x,y),z) -> div(x,times(y,z))
  eq(|0|(),|0|()) -> true()
  eq(s(x),|0|()) -> false()
  eq(|0|(),s(y)) -> false()
  eq(s(x),s(y)) -> eq(x,y)
  divides(y,x) -> eq(x,times(div(x,y),y))
  prime(s(s(x))) -> pr(s(s(x)),s(x))
  pr(x,s(|0|())) -> true()
  pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
  if(true(),x,y) -> false()
  if(false(),x,y) -> pr(x,y)


-- SCC decomposition.

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

p1: plus#(s(x),y) -> plus#(x,y)
p2: times#(s(x),y) -> plus#(y,times(x,y))
p3: times#(s(x),y) -> times#(x,y)
p4: div#(x,y) -> quot#(x,y,y)
p5: quot#(s(x),s(y),z) -> quot#(x,y,z)
p6: quot#(x,|0|(),s(z)) -> div#(x,s(z))
p7: div#(div(x,y),z) -> div#(x,times(y,z))
p8: div#(div(x,y),z) -> times#(y,z)
p9: eq#(s(x),s(y)) -> eq#(x,y)
p10: divides#(y,x) -> eq#(x,times(div(x,y),y))
p11: divides#(y,x) -> times#(div(x,y),y)
p12: divides#(y,x) -> div#(x,y)
p13: prime#(s(s(x))) -> pr#(s(s(x)),s(x))
p14: pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
p15: pr#(x,s(s(y))) -> divides#(s(s(y)),x)
p16: if#(false(),x,y) -> pr#(x,y)

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(x,y)

The estimated dependency graph contains the following SCCs:

  {p14, p16}
  {p4, p5, p6, p7}
  {p3}
  {p1}
  {p9}


-- Reduction pair.

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

p1: if#(false(),x,y) -> pr#(x,y)
p2: pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(x,y)

The set of usable rules consists of

  r4

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if#_A(x1,x2,x3) = max{28, x3 + 10}
        false_A = 20
        pr#_A(x1,x2) = x2 + 2
        s_A(x1) = max{17, x1 + 9}
        divides_A(x1,x2) = max{25, x1 - 4}
        times_A(x1,x2) = max{26, x1 - 22, x2 + 9}
        |0|_A = 21
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            if#_A(x1,x2,x3) = x3
            false_A() = 13
            pr#_A(x1,x2) = x2
            s_A(x1) = x1 + 3
            divides_A(x1,x2) = 1
            times_A(x1,x2) = x1 + 2
            |0|_A() = 9
            if#!_A(x1,x2,x3) = x2 + 5
            false!_A() = 5
            pr#!_A(x1,x2) = x1 + x2
            s!_A(x1) = x1 + 7
            divides!_A(x1,x2) = 6
            times!_A(x1,x2) = x1 + x2 + 8
            |0|!_A() = 12
  
    partial status:
  
      pi(if#) = []
      pi(false) = []
      pi(pr#) = [2]
      pi(s) = [1]
      pi(divides) = []
      pi(times) = [2]
      pi(|0|) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

p1: div#(div(x,y),z) -> div#(x,times(y,z))
p2: div#(x,y) -> quot#(x,y,y)
p3: quot#(x,|0|(),s(z)) -> div#(x,s(z))
p4: quot#(s(x),s(y),z) -> quot#(x,y,z)

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(x,y)

The set of usable rules consists of

  r1, r2, r3, r4, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        div#_A(x1,x2) = x1
        div_A(x1,x2) = max{x1 + 6, x2 - 2}
        times_A(x1,x2) = max{15, x1 + 7, x2 + 1}
        quot#_A(x1,x2,x3) = x1
        |0|_A = 0
        s_A(x1) = x1
        plus_A(x1,x2) = max{16, x1 + 2, x2 + 1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            div#_A(x1,x2) = 0
            div_A(x1,x2) = x1 + x2 + 2
            times_A(x1,x2) = x1 + 1
            quot#_A(x1,x2,x3) = 0
            |0|_A() = 5
            s_A(x1) = x1 + 1
            plus_A(x1,x2) = x1 + x2 + 2
            div#!_A(x1,x2) = x1 + 3
            div!_A(x1,x2) = x1 + x2 + 8
            times!_A(x1,x2) = x1 + x2 + 11
            quot#!_A(x1,x2,x3) = x1 + 3
            |0|!_A() = 2
            s!_A(x1) = x1 + 4
            plus!_A(x1,x2) = x1 + x2 + 6
  
    partial status:
  
      pi(div#) = [1]
      pi(div) = [1]
      pi(times) = []
      pi(quot#) = [1]
      pi(|0|) = []
      pi(s) = [1]
      pi(plus) = []

The following rules are removed:

  p1, p4



-- SCC decomposition.

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

p1: div#(x,y) -> quot#(x,y,y)
p2: quot#(x,|0|(),s(z)) -> div#(x,s(z))

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(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: div#(x,y) -> quot#(x,y,y)
p2: quot#(x,|0|(),s(z)) -> div#(x,s(z))

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(x,y)

The set of usable rules consists of

  r7

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        div#_A(x1,x2) = 1
        quot#_A(x1,x2,x3) = 1
        |0|_A = 18
        s_A(x1) = max{16, x1 + 6}
        div_A(x1,x2) = max{9, x1 + 6, x2 - 2}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            div#_A(x1,x2) = x2
            quot#_A(x1,x2,x3) = x3
            |0|_A() = 6
            s_A(x1) = 4
            div_A(x1,x2) = x1 + x2 + 1
            div#!_A(x1,x2) = x2 + 1
            quot#!_A(x1,x2,x3) = x2
            |0|!_A() = 22
            s!_A(x1) = 7
            div!_A(x1,x2) = 18
  
    partial status:
  
      pi(div#) = []
      pi(quot#) = []
      pi(|0|) = []
      pi(s) = [1]
      pi(div) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(x,y)

The set of usable rules consists of

  r4, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        times#_A(x1,x2) = max{x1 + 16, x2 + 19}
        s_A(x1) = max{17, x1 + 15}
        times_A(x1,x2) = max{31, x1 - 5, x2 + 15}
        |0|_A = 6
        div_A(x1,x2) = x1 + 15
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            times#_A(x1,x2) = x1
            s_A(x1) = x1 + 6
            times_A(x1,x2) = x1 + 8
            |0|_A() = 9
            div_A(x1,x2) = x1 + x2 + 7
            times#!_A(x1,x2) = x1
            s!_A(x1) = 13
            times!_A(x1,x2) = x2 + 10
            |0|!_A() = 12
            div!_A(x1,x2) = x1 + x2 + 3
  
    partial status:
  
      pi(times#) = [1, 2]
      pi(s) = [1]
      pi(times) = []
      pi(|0|) = []
      pi(div) = [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(x),y) -> plus#(x,y)

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(x,y)

The set of usable rules consists of

  r4, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = x1
        s_A(x1) = x1
        times_A(x1,x2) = max{5, x1 - 6, x2 + 2}
        |0|_A = 0
        div_A(x1,x2) = x1 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            plus#_A(x1,x2) = x1
            s_A(x1) = x1 + 1
            times_A(x1,x2) = x1 + 2
            |0|_A() = 9
            div_A(x1,x2) = x1 + x2 + 1
            plus#!_A(x1,x2) = 0
            s!_A(x1) = 3
            times!_A(x1,x2) = 2
            |0|!_A() = 8
            div!_A(x1,x2) = x1 + x2
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = [1]
      pi(times) = []
      pi(|0|) = []
      pi(div) = [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: eq#(s(x),s(y)) -> eq#(x,y)

and R consists of:

r1: plus(x,|0|()) -> x
r2: plus(|0|(),y) -> y
r3: plus(s(x),y) -> s(plus(x,y))
r4: times(|0|(),y) -> |0|()
r5: times(s(|0|()),y) -> y
r6: times(s(x),y) -> plus(y,times(x,y))
r7: div(|0|(),y) -> |0|()
r8: div(x,y) -> quot(x,y,y)
r9: quot(|0|(),s(y),z) -> |0|()
r10: quot(s(x),s(y),z) -> quot(x,y,z)
r11: quot(x,|0|(),s(z)) -> s(div(x,s(z)))
r12: div(div(x,y),z) -> div(x,times(y,z))
r13: eq(|0|(),|0|()) -> true()
r14: eq(s(x),|0|()) -> false()
r15: eq(|0|(),s(y)) -> false()
r16: eq(s(x),s(y)) -> eq(x,y)
r17: divides(y,x) -> eq(x,times(div(x,y),y))
r18: prime(s(s(x))) -> pr(s(s(x)),s(x))
r19: pr(x,s(|0|())) -> true()
r20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
r21: if(true(),x,y) -> false()
r22: if(false(),x,y) -> pr(x,y)

The set of usable rules consists of

  r4, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        eq#_A(x1,x2) = max{6, x1 - 1}
        s_A(x1) = max{16, x1 + 5}
        times_A(x1,x2) = max{23, x1 - 2, x2 + 3}
        |0|_A = 8
        div_A(x1,x2) = 15
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            eq#_A(x1,x2) = x1 + x2
            s_A(x1) = x1 + 2
            times_A(x1,x2) = x1 + 3
            |0|_A() = 12
            div_A(x1,x2) = x1 + 4
            eq#!_A(x1,x2) = x1
            s!_A(x1) = x1 + 7
            times!_A(x1,x2) = 11
            |0|!_A() = 10
            div!_A(x1,x2) = 9
  
    partial status:
  
      pi(eq#) = []
      pi(s) = []
      pi(times) = []
      pi(|0|) = []
      pi(div) = []

The following rules are removed:

  p1

Then no dependency pair remains.