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:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_active#_A(x1,x2,x3) = 12
        false_A = 1
        mark#_A(x1) = 12
        if_A(x1,x2,x3) = max{x2 + 13, x3 + 15}
        mark_A(x1) = max{11, x1 + 8}
        true_A = 1
        div_A(x1,x2) = 10
        div_active#_A(x1,x2) = 12
        s_A(x1) = 2
        ge_active_A(x1,x2) = 13
        minus_A(x1,x2) = 1
        |0|_A = 0
        minus_active_A(x1,x2) = 10
        ge_A(x1,x2) = 6
        div_active_A(x1,x2) = 17
        if_active_A(x1,x2,x3) = max{x2 + 14, x3 + 15}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_active#_A(x1,x2,x3) = max{0, x1 - 40, x2 - 50, x3 - 30}
          false_A = 52
          mark#_A(x1) = max{1, x1 - 50}
          if_A(x1,x2,x3) = max{x1 + 49, x2 + 39, x3 + 39}
          mark_A(x1) = x1 + 39
          true_A = 53
          div_A(x1,x2) = max{51, x1 + 38, x2 + 34}
          div_active#_A(x1,x2) = max{0, x2 - 16}
          s_A(x1) = max{33, x1}
          ge_active_A(x1,x2) = max{49, x2 + 24}
          minus_A(x1,x2) = 1
          |0|_A = 29
          minus_active_A(x1,x2) = 38
          ge_A(x1,x2) = max{25, x2 - 7}
          div_active_A(x1,x2) = max{x1 + 38, x2 + 73}
          if_active_A(x1,x2,x3) = max{x1 + 49, x2 + 39, x3 + 45}
          if_active#!_A(x1,x2,x3) = max{x1 + 30, x2 + 44, x3 + 82}
          false!_A = 33
          mark#!_A(x1) = max{81, x1 + 44}
          if!_A(x1,x2,x3) = 66
          mark!_A(x1) = max{171, x1 + 120}
          true!_A = 113
          div!_A(x1,x2) = max{x1 + 228, x2 - 72}
          div_active#!_A(x1,x2) = max{79, x2 + 78}
          s!_A(x1) = 230
          ge_active!_A(x1,x2) = max{112, x2 - 1}
          minus!_A(x1,x2) = max{0, x1 - 148}
          |0|!_A = 77
          minus_active!_A(x1,x2) = max{0, x1 - 148}
          ge!_A(x1,x2) = 1
          div_active!_A(x1,x2) = max{227, x1 - 38, x2 - 340}
          if_active!_A(x1,x2,x3) = max{170, x1 + 118, x2 + 159}
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(if) = [2]
      pi(mark) = [1]
      pi(true) = []
      pi(div) = []
      pi(div_active#) = []
      pi(s) = []
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(minus_active) = []
      pi(ge) = []
      pi(div_active) = []
      pi(if_active) = [2]

The following rules are removed:

  p1, p2, p3, p5



-- SCC decomposition.

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

p1: if_active#(true(),x,y) -> mark#(x)
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: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: if_active#(true(),x,y) -> mark#(x)
p2: mark#(s(x)) -> mark#(x)
p3: mark#(div(x,y)) -> div_active#(mark(x),y)
p4: 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 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:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_active#_A(x1,x2,x3) = 5
        true_A = 2
        mark#_A(x1) = 5
        s_A(x1) = max{9, x1}
        div_A(x1,x2) = max{41, x1 - 8, x2 + 10}
        div_active#_A(x1,x2) = 5
        mark_A(x1) = max{7, x1}
        ge_active_A(x1,x2) = max{4, x1 + 1, x2 - 19}
        minus_A(x1,x2) = x1
        |0|_A = 20
        minus_active_A(x1,x2) = max{7, x1}
        false_A = 6
        ge_A(x1,x2) = max{4, x1 + 1, x2 - 19}
        div_active_A(x1,x2) = max{41, x1 - 8, x2 + 10}
        if_A(x1,x2,x3) = max{x2, x3 + 20}
        if_active_A(x1,x2,x3) = max{x2, x3 + 20}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_active#_A(x1,x2,x3) = max{74, x2 - 110, x3 - 13}
          true_A = 3
          mark#_A(x1) = max{74, x1 - 110}
          s_A(x1) = max{84, x1}
          div_A(x1,x2) = max{133, x1 + 107}
          div_active#_A(x1,x2) = max{74, x1 - 11}
          mark_A(x1) = max{11, x1 + 8}
          ge_active_A(x1,x2) = max{3, x2 - 3}
          minus_A(x1,x2) = 4
          |0|_A = 2
          minus_active_A(x1,x2) = 12
          false_A = 0
          ge_A(x1,x2) = max{3, x2 - 7}
          div_active_A(x1,x2) = max{141, x1 + 107}
          if_A(x1,x2,x3) = max{141, x2 + 58, x3 + 132}
          if_active_A(x1,x2,x3) = max{141, x2 + 58, x3 + 140}
          if_active#!_A(x1,x2,x3) = max{x2 - 94, x3 + 18}
          true!_A = 2
          mark#!_A(x1) = max{1, x1 - 106}
          s!_A(x1) = 18
          div!_A(x1,x2) = x2
          div_active#!_A(x1,x2) = max{19, x1 - 9}
          mark!_A(x1) = max{0, x1 - 112}
          ge_active!_A(x1,x2) = max{85, x1 - 85, x2 - 85}
          minus!_A(x1,x2) = 0
          |0|!_A = 0
          minus_active!_A(x1,x2) = 0
          false!_A = 18
          ge!_A(x1,x2) = max{85, x1 - 85, x2 - 85}
          div_active!_A(x1,x2) = x2
          if!_A(x1,x2,x3) = 0
          if_active!_A(x1,x2,x3) = max{19, x2 - 111}
  
    partial status:
  
      pi(if_active#) = []
      pi(true) = []
      pi(mark#) = []
      pi(s) = []
      pi(div) = []
      pi(div_active#) = []
      pi(mark) = [1]
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(minus_active) = []
      pi(false) = []
      pi(ge) = []
      pi(div_active) = []
      pi(if) = [2]
      pi(if_active) = [2, 3]

The following rules are removed:

  p1, p3, p4



-- SCC decomposition.

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

p1: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mark#_A(x1) = max{42, x1 + 11}
        s_A(x1) = x1 + 42
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          mark#_A(x1) = 0
          s_A(x1) = max{3, x1 - 3}
          mark#!_A(x1) = 0
          s!_A(x1) = max{0, x1 - 80}
  
    partial status:
  
      pi(mark#) = [1]
      pi(s) = [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: 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 reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_active#_A(x1,x2) = max{11, x2}
        s_A(x1) = x1 + 11
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus_active#_A(x1,x2) = 0
          s_A(x1) = max{10, x1 - 3}
          minus_active#!_A(x1,x2) = x2 + 5
          s!_A(x1) = max{9, x1 + 6}
  
    partial status:
  
      pi(minus_active#) = [2]
      pi(s) = [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: 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:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ge_active#_A(x1,x2) = max{28, x2}
        s_A(x1) = x1 + 28
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          ge_active#_A(x1,x2) = 0
          s_A(x1) = max{14, x1 - 3}
          ge_active#!_A(x1,x2) = x2 + 5
          s!_A(x1) = max{9, x1 + 6}
  
    partial status:
  
      pi(ge_active#) = [2]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.