YES

We show the termination of the TRS R:

  minus_active(|0|(),y) -> |0|()
  mark(|0|()) -> |0|()
  minus_active(s(x),s(y)) -> minus_active(x,y)
  mark(s(x)) -> s(mark(x))
  ge_active(x,|0|()) -> true()
  mark(minus(x,y)) -> minus_active(x,y)
  ge_active(|0|(),s(y)) -> false()
  mark(ge(x,y)) -> ge_active(x,y)
  ge_active(s(x),s(y)) -> ge_active(x,y)
  mark(div(x,y)) -> div_active(mark(x),y)
  div_active(|0|(),s(y)) -> |0|()
  mark(if(x,y,z)) -> if_active(mark(x),y,z)
  div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
  if_active(true(),x,y) -> mark(x)
  minus_active(x,y) -> minus(x,y)
  if_active(false(),x,y) -> mark(y)
  ge_active(x,y) -> ge(x,y)
  if_active(x,y,z) -> if(x,y,z)
  div_active(x,y) -> div(x,y)

-- SCC decomposition.

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

p1: minus_active#(s(x),s(y)) -> minus_active#(x,y)
p2: mark#(s(x)) -> mark#(x)
p3: mark#(minus(x,y)) -> minus_active#(x,y)
p4: mark#(ge(x,y)) -> ge_active#(x,y)
p5: ge_active#(s(x),s(y)) -> ge_active#(x,y)
p6: mark#(div(x,y)) -> div_active#(mark(x),y)
p7: mark#(div(x,y)) -> mark#(x)
p8: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)
p9: mark#(if(x,y,z)) -> mark#(x)
p10: div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
p11: div_active#(s(x),s(y)) -> ge_active#(x,y)
p12: if_active#(true(),x,y) -> mark#(x)
p13: if_active#(false(),x,y) -> mark#(y)

and R consists of:

r1: minus_active(|0|(),y) -> |0|()
r2: mark(|0|()) -> |0|()
r3: minus_active(s(x),s(y)) -> minus_active(x,y)
r4: mark(s(x)) -> s(mark(x))
r5: ge_active(x,|0|()) -> true()
r6: mark(minus(x,y)) -> minus_active(x,y)
r7: ge_active(|0|(),s(y)) -> false()
r8: mark(ge(x,y)) -> ge_active(x,y)
r9: ge_active(s(x),s(y)) -> ge_active(x,y)
r10: mark(div(x,y)) -> div_active(mark(x),y)
r11: div_active(|0|(),s(y)) -> |0|()
r12: mark(if(x,y,z)) -> if_active(mark(x),y,z)
r13: div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
r14: if_active(true(),x,y) -> mark(x)
r15: minus_active(x,y) -> minus(x,y)
r16: if_active(false(),x,y) -> mark(y)
r17: ge_active(x,y) -> ge(x,y)
r18: if_active(x,y,z) -> if(x,y,z)
r19: div_active(x,y) -> div(x,y)

The estimated dependency graph contains the following SCCs:

  {p2, p6, p7, p8, p9, p10, p12, p13}
  {p1}
  {p5}


-- Reduction pair.

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

p1: if_active#(false(),x,y) -> mark#(y)
p2: mark#(if(x,y,z)) -> mark#(x)
p3: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)
p4: if_active#(true(),x,y) -> mark#(x)
p5: mark#(div(x,y)) -> mark#(x)
p6: mark#(div(x,y)) -> div_active#(mark(x),y)
p7: div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
p8: mark#(s(x)) -> mark#(x)

and R consists of:

r1: minus_active(|0|(),y) -> |0|()
r2: mark(|0|()) -> |0|()
r3: minus_active(s(x),s(y)) -> minus_active(x,y)
r4: mark(s(x)) -> s(mark(x))
r5: ge_active(x,|0|()) -> true()
r6: mark(minus(x,y)) -> minus_active(x,y)
r7: ge_active(|0|(),s(y)) -> false()
r8: mark(ge(x,y)) -> ge_active(x,y)
r9: ge_active(s(x),s(y)) -> ge_active(x,y)
r10: mark(div(x,y)) -> div_active(mark(x),y)
r11: div_active(|0|(),s(y)) -> |0|()
r12: mark(if(x,y,z)) -> if_active(mark(x),y,z)
r13: div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
r14: if_active(true(),x,y) -> mark(x)
r15: minus_active(x,y) -> minus(x,y)
r16: if_active(false(),x,y) -> mark(y)
r17: ge_active(x,y) -> ge(x,y)
r18: if_active(x,y,z) -> if(x,y,z)
r19: div_active(x,y) -> div(x,y)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        if_active#_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((0,1),(0,0)) x3
        false_A() = (1,0)
        mark#_A(x1) = ((0,1),(0,0)) x1
        if_A(x1,x2,x3) = x1 + x2 + x3 + (2,0)
        mark_A(x1) = x1 + (2,0)
        true_A() = (1,0)
        div_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,0)) x2 + (1,1)
        div_active#_A(x1,x2) = ((0,1),(0,0)) x1 + (1,0)
        s_A(x1) = x1 + (1,5)
        ge_active_A(x1,x2) = (2,0)
        minus_A(x1,x2) = x1 + (1,0)
        |0|_A() = (1,0)
        minus_active_A(x1,x2) = x1 + (2,0)
        div_active_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,0)) x2 + (2,1)
        if_active_A(x1,x2,x3) = x1 + x2 + x3 + (3,0)
        ge_A(x1,x2) = (1,0)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        if_active#_A(x1,x2,x3) = (0,0)
        false_A() = (1,3)
        mark#_A(x1) = (0,0)
        if_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (12,14)
        mark_A(x1) = ((0,0),(1,0)) x1 + (2,0)
        true_A() = (1,1)
        div_A(x1,x2) = (1,1)
        div_active#_A(x1,x2) = (0,0)
        s_A(x1) = (1,1)
        ge_active_A(x1,x2) = (3,2)
        minus_A(x1,x2) = ((1,1),(0,1)) x1 + (4,6)
        |0|_A() = (10,6)
        minus_active_A(x1,x2) = ((0,1),(0,0)) x1 + (3,5)
        div_active_A(x1,x2) = (11,1)
        if_active_A(x1,x2,x3) = ((1,0),(1,0)) x2 + (11,13)
        ge_A(x1,x2) = (1,3)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        if_active#_A(x1,x2,x3) = (0,0)
        false_A() = (1,1)
        mark#_A(x1) = (0,0)
        if_A(x1,x2,x3) = ((0,1),(0,0)) x2 + (1,3)
        mark_A(x1) = (2,1)
        true_A() = (4,3)
        div_A(x1,x2) = (4,3)
        div_active#_A(x1,x2) = (0,0)
        s_A(x1) = (1,1)
        ge_active_A(x1,x2) = (3,2)
        minus_A(x1,x2) = (4,3)
        |0|_A() = (1,3)
        minus_active_A(x1,x2) = (3,2)
        div_active_A(x1,x2) = (3,2)
        if_active_A(x1,x2,x3) = x2 + (4,2)
        ge_A(x1,x2) = (1,1)
    

The next rules are strictly ordered:

  p5, p8

We remove them from the problem.

-- SCC decomposition.

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

p1: if_active#(false(),x,y) -> mark#(y)
p2: mark#(if(x,y,z)) -> mark#(x)
p3: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)
p4: if_active#(true(),x,y) -> mark#(x)
p5: mark#(div(x,y)) -> div_active#(mark(x),y)
p6: div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())

and R consists of:

r1: minus_active(|0|(),y) -> |0|()
r2: mark(|0|()) -> |0|()
r3: minus_active(s(x),s(y)) -> minus_active(x,y)
r4: mark(s(x)) -> s(mark(x))
r5: ge_active(x,|0|()) -> true()
r6: mark(minus(x,y)) -> minus_active(x,y)
r7: ge_active(|0|(),s(y)) -> false()
r8: mark(ge(x,y)) -> ge_active(x,y)
r9: ge_active(s(x),s(y)) -> ge_active(x,y)
r10: mark(div(x,y)) -> div_active(mark(x),y)
r11: div_active(|0|(),s(y)) -> |0|()
r12: mark(if(x,y,z)) -> if_active(mark(x),y,z)
r13: div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
r14: if_active(true(),x,y) -> mark(x)
r15: minus_active(x,y) -> minus(x,y)
r16: if_active(false(),x,y) -> mark(y)
r17: ge_active(x,y) -> ge(x,y)
r18: if_active(x,y,z) -> if(x,y,z)
r19: div_active(x,y) -> div(x,y)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: if_active#(false(),x,y) -> mark#(y)
p2: mark#(div(x,y)) -> div_active#(mark(x),y)
p3: div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
p4: if_active#(true(),x,y) -> mark#(x)
p5: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)
p6: mark#(if(x,y,z)) -> mark#(x)

and R consists of:

r1: minus_active(|0|(),y) -> |0|()
r2: mark(|0|()) -> |0|()
r3: minus_active(s(x),s(y)) -> minus_active(x,y)
r4: mark(s(x)) -> s(mark(x))
r5: ge_active(x,|0|()) -> true()
r6: mark(minus(x,y)) -> minus_active(x,y)
r7: ge_active(|0|(),s(y)) -> false()
r8: mark(ge(x,y)) -> ge_active(x,y)
r9: ge_active(s(x),s(y)) -> ge_active(x,y)
r10: mark(div(x,y)) -> div_active(mark(x),y)
r11: div_active(|0|(),s(y)) -> |0|()
r12: mark(if(x,y,z)) -> if_active(mark(x),y,z)
r13: div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
r14: if_active(true(),x,y) -> mark(x)
r15: minus_active(x,y) -> minus(x,y)
r16: if_active(false(),x,y) -> mark(y)
r17: ge_active(x,y) -> ge(x,y)
r18: if_active(x,y,z) -> if(x,y,z)
r19: div_active(x,y) -> div(x,y)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        if_active#_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (0,2)
        false_A() = (1,1)
        mark#_A(x1) = ((0,1),(1,1)) x1
        div_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (4,5)
        div_active#_A(x1,x2) = x1 + x2 + (3,9)
        mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
        s_A(x1) = (1,1)
        ge_active_A(x1,x2) = (2,1)
        minus_A(x1,x2) = (1,1)
        |0|_A() = (0,1)
        true_A() = (1,1)
        if_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,3)
        minus_active_A(x1,x2) = (2,1)
        div_active_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (5,8)
        if_active_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (2,3)
        ge_A(x1,x2) = (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        if_active#_A(x1,x2,x3) = ((0,0),(1,1)) x1 + ((0,1),(0,0)) x3
        false_A() = (3,3)
        mark#_A(x1) = (0,7)
        div_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,1)) x2 + (1,1)
        div_active#_A(x1,x2) = ((1,0),(1,0)) x1 + x2
        mark_A(x1) = ((0,1),(0,0)) x1 + (8,1)
        s_A(x1) = (1,2)
        ge_active_A(x1,x2) = (2,2)
        minus_A(x1,x2) = (1,3)
        |0|_A() = (14,3)
        true_A() = (1,3)
        if_A(x1,x2,x3) = x1 + x3 + (5,3)
        minus_active_A(x1,x2) = (12,2)
        div_active_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,0),(1,0)) x2 + (9,0)
        if_active_A(x1,x2,x3) = x1 + x2 + x3 + (4,2)
        ge_A(x1,x2) = (3,3)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        if_active#_A(x1,x2,x3) = (1,8)
        false_A() = (1,12)
        mark#_A(x1) = (2,9)
        div_A(x1,x2) = ((1,1),(0,0)) x1 + (5,12)
        div_active#_A(x1,x2) = x1 + ((0,0),(1,0)) x2
        mark_A(x1) = (3,10)
        s_A(x1) = (4,3)
        ge_active_A(x1,x2) = (4,11)
        minus_A(x1,x2) = (1,1)
        |0|_A() = (5,16)
        true_A() = (1,1)
        if_A(x1,x2,x3) = x1 + (1,9)
        minus_active_A(x1,x2) = (4,11)
        div_active_A(x1,x2) = ((0,0),(1,0)) x2 + (4,11)
        if_active_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,1),(0,0)) x2 + ((0,0),(1,0)) x3 + (2,8)
        ge_A(x1,x2) = (5,12)
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6

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_active#(s(x),s(y)) -> minus_active#(x,y)

and R consists of:

r1: minus_active(|0|(),y) -> |0|()
r2: mark(|0|()) -> |0|()
r3: minus_active(s(x),s(y)) -> minus_active(x,y)
r4: mark(s(x)) -> s(mark(x))
r5: ge_active(x,|0|()) -> true()
r6: mark(minus(x,y)) -> minus_active(x,y)
r7: ge_active(|0|(),s(y)) -> false()
r8: mark(ge(x,y)) -> ge_active(x,y)
r9: ge_active(s(x),s(y)) -> ge_active(x,y)
r10: mark(div(x,y)) -> div_active(mark(x),y)
r11: div_active(|0|(),s(y)) -> |0|()
r12: mark(if(x,y,z)) -> if_active(mark(x),y,z)
r13: div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
r14: if_active(true(),x,y) -> mark(x)
r15: minus_active(x,y) -> minus(x,y)
r16: if_active(false(),x,y) -> mark(y)
r17: ge_active(x,y) -> ge(x,y)
r18: if_active(x,y,z) -> if(x,y,z)
r19: div_active(x,y) -> div(x,y)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

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

The next rules are strictly ordered:

  p1
  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19

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

and R consists of:

r1: minus_active(|0|(),y) -> |0|()
r2: mark(|0|()) -> |0|()
r3: minus_active(s(x),s(y)) -> minus_active(x,y)
r4: mark(s(x)) -> s(mark(x))
r5: ge_active(x,|0|()) -> true()
r6: mark(minus(x,y)) -> minus_active(x,y)
r7: ge_active(|0|(),s(y)) -> false()
r8: mark(ge(x,y)) -> ge_active(x,y)
r9: ge_active(s(x),s(y)) -> ge_active(x,y)
r10: mark(div(x,y)) -> div_active(mark(x),y)
r11: div_active(|0|(),s(y)) -> |0|()
r12: mark(if(x,y,z)) -> if_active(mark(x),y,z)
r13: div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
r14: if_active(true(),x,y) -> mark(x)
r15: minus_active(x,y) -> minus(x,y)
r16: if_active(false(),x,y) -> mark(y)
r17: ge_active(x,y) -> ge(x,y)
r18: if_active(x,y,z) -> if(x,y,z)
r19: div_active(x,y) -> div(x,y)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

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

The next rules are strictly ordered:

  p1

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