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

  r2, r5, r7, r11

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_active#_A(x1,x2,x3) = x2 + x3 + 7
          false_A() = 3
          mark#_A(x1) = x1 + 7
          if_A(x1,x2,x3) = x1 + x2 + x3 + 5
          mark_A(x1) = x1 + 2
          true_A() = 1
          div_A(x1,x2) = x1 + 6
          div_active#_A(x1,x2) = 13
          s_A(x1) = x1
          ge_active_A(x1,x2) = 4
          minus_A(x1,x2) = 0
          |0|_A() = 0
          div_active_A(x1,x2) = 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_active#_A(x1,x2,x3) = 1
          false_A = 66
          mark#_A(x1) = 1
          if_A(x1,x2,x3) = max{70, x3 + 68}
          mark_A(x1) = x1 + 4
          true_A = 4
          div_A(x1,x2) = x2 + 32
          div_active#_A(x1,x2) = 1
          s_A(x1) = max{31, x1}
          ge_active_A(x1,x2) = max{x1 + 4, x2 + 66}
          minus_A(x1,x2) = max{13, x1 - 5}
          |0|_A = 19
          div_active_A(x1,x2) = max{75, x1 - 31}
          if_active#!_A(x1,x2,x3) = max{54, x2 - 8, x3 + 2}
          false!_A = 26
          mark#!_A(x1) = max{24, x1 - 8}
          if!_A(x1,x2,x3) = max{x1 - 9, x2}
          mark!_A(x1) = max{0, x1 - 65}
          true!_A = 10
          div!_A(x1,x2) = 21
          div_active#!_A(x1,x2) = x2 + 24
          s!_A(x1) = 25
          ge_active!_A(x1,x2) = 21
          minus!_A(x1,x2) = max{26, x1 - 15, x2 - 1}
          |0|!_A = 9
          div_active!_A(x1,x2) = 21
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(if) = [1, 3]
      pi(mark) = [1]
      pi(true) = []
      pi(div) = []
      pi(div_active#) = []
      pi(s) = [1]
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(div_active) = []

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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_active#_A(x1,x2,x3) = x1 + x2 + x3 + 1
          true_A() = 2
          mark#_A(x1) = x1 + 3
          s_A(x1) = x1 + 4
          div_A(x1,x2) = x1 + x2 + 1
          div_active#_A(x1,x2) = x1 + x2 + 4
          mark_A(x1) = x1
          ge_active_A(x1,x2) = 2
          minus_A(x1,x2) = x1
          |0|_A() = 0
          minus_active_A(x1,x2) = x1
          false_A() = 1
          ge_A(x1,x2) = 2
          div_active_A(x1,x2) = x1 + x2 + 1
          if_A(x1,x2,x3) = x2 + x3
          if_active_A(x1,x2,x3) = x2 + x3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_active#_A(x1,x2,x3) = max{1, x3 - 26}
          true_A = 1
          mark#_A(x1) = 1
          s_A(x1) = max{9, x1 - 25}
          div_A(x1,x2) = max{x1 + 76, x2 + 62}
          div_active#_A(x1,x2) = 1
          mark_A(x1) = x1 + 32
          ge_active_A(x1,x2) = 9
          minus_A(x1,x2) = 2
          |0|_A = 27
          minus_active_A(x1,x2) = 27
          false_A = 1
          ge_A(x1,x2) = 1
          div_active_A(x1,x2) = max{x1 + 76, x2 + 69}
          if_A(x1,x2,x3) = max{53, x1 - 10, x2, x3 + 52}
          if_active_A(x1,x2,x3) = max{85, x1 - 10, x2 + 32, x3 + 52}
          if_active#!_A(x1,x2,x3) = max{15, x2 + 4, x3 + 4}
          true!_A = 0
          mark#!_A(x1) = max{3, x1 - 6}
          s!_A(x1) = 1
          div!_A(x1,x2) = 2
          div_active#!_A(x1,x2) = x2 + 54
          mark!_A(x1) = 2
          ge_active!_A(x1,x2) = 0
          minus!_A(x1,x2) = 0
          |0|!_A = 0
          minus_active!_A(x1,x2) = 0
          false!_A = 2
          ge!_A(x1,x2) = 0
          div_active!_A(x1,x2) = 2
          if!_A(x1,x2,x3) = 0
          if_active!_A(x1,x2,x3) = 2
  
    partial status:
  
      pi(if_active#) = [2, 3]
      pi(true) = []
      pi(mark#) = [1]
      pi(s) = []
      pi(div) = []
      pi(div_active#) = []
      pi(mark) = []
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(minus_active) = []
      pi(false) = []
      pi(ge) = []
      pi(div_active) = []
      pi(if) = []
      pi(if_active) = []

The following rules are removed:

  p1, p2, p3, p4

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

  r1, r2, r11

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_active#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 3
          minus_active_A(x1,x2) = x1 + x2 + 10
          |0|_A() = 2
          mark_A(x1) = x1 + 4
          div_active_A(x1,x2) = x2 + 1
  
    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{3, x1 - 3}
          minus_active_A(x1,x2) = max{3, x1 - 1, x2 - 1}
          |0|_A = 0
          mark_A(x1) = max{2, x1 - 48}
          div_active_A(x1,x2) = max{0, x1 - 57, x2 - 57}
          minus_active#!_A(x1,x2) = max{8, x1 - 4, x2 - 4}
          s!_A(x1) = max{7, x1 - 54}
          minus_active!_A(x1,x2) = max{10, x1 - 55, x2 - 4}
          |0|!_A = 6
          mark!_A(x1) = max{8, x1 - 35}
          div_active!_A(x1,x2) = max{8, x1 - 35, x2 - 21}
  
    partial status:
  
      pi(minus_active#) = []
      pi(s) = [1]
      pi(minus_active) = [2]
      pi(|0|) = []
      pi(mark) = []
      pi(div_active) = [2]

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

  r1, r2, r11

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge_active#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 3
          minus_active_A(x1,x2) = x1 + x2 + 14
          |0|_A() = 2
          mark_A(x1) = x1 + 9
          div_active_A(x1,x2) = x2 + 9
  
    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{3, x1 - 3}
          minus_active_A(x1,x2) = max{3, x1 - 1, x2 - 1}
          |0|_A = 0
          mark_A(x1) = max{2, x1 - 28}
          div_active_A(x1,x2) = max{0, x1 - 37, x2 - 37}
          ge_active#!_A(x1,x2) = max{7, x1 - 4, x2 - 4}
          s!_A(x1) = max{6, x1 - 56}
          minus_active!_A(x1,x2) = max{5, x1 - 35, x2 - 4}
          |0|!_A = 5
          mark!_A(x1) = max{9, x1 - 34}
          div_active!_A(x1,x2) = max{8, x1 - 28, x2 - 36}
  
    partial status:
  
      pi(ge_active#) = []
      pi(s) = [1]
      pi(minus_active) = [2]
      pi(|0|) = []
      pi(mark) = []
      pi(div_active) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.