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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              if_active#_A(x1,x2,x3) = 0
              false_A() = 1
              mark#_A(x1) = 0
              if_A(x1,x2,x3) = x2 + x3
              mark_A(x1) = x1
              true_A() = 1
              div_A(x1,x2) = x1 + x2
              div_active#_A(x1,x2) = 0
              s_A(x1) = x1 + 1
              ge_active_A(x1,x2) = 1
              minus_A(x1,x2) = 0
              |0|_A() = 0
              minus_active_A(x1,x2) = 0
              ge_A(x1,x2) = 1
              div_active_A(x1,x2) = x1 + x2
              if_active_A(x1,x2,x3) = x2 + x3
              if_active#!_A(x1,x2,x3) = 2
              false!_A() = 1
              mark#!_A(x1) = 2
              if!_A(x1,x2,x3) = 4
              mark!_A(x1) = 5
              true!_A() = 3
              div!_A(x1,x2) = 5
              div_active#!_A(x1,x2) = 2
              s!_A(x1) = 5
              ge_active!_A(x1,x2) = 4
              minus!_A(x1,x2) = 1
              |0|!_A() = 0
              minus_active!_A(x1,x2) = 4
              ge!_A(x1,x2) = 4
              div_active!_A(x1,x2) = 5
              if_active!_A(x1,x2,x3) = 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_active#_A(x1,x2,x3) = 1
            false_A = 5
            mark#_A(x1) = 1
            if_A(x1,x2,x3) = max{146, x1 + 81, x2 + 41, x3 + 81}
            mark_A(x1) = x1 + 3
            true_A = 1
            div_A(x1,x2) = x1 + 89
            div_active#_A(x1,x2) = 1
            s_A(x1) = max{57, x1 + 11}
            ge_active_A(x1,x2) = 5
            minus_A(x1,x2) = 1
            |0|_A = 4
            minus_active_A(x1,x2) = 4
            ge_A(x1,x2) = 2
            div_active_A(x1,x2) = x1 + 89
            if_active_A(x1,x2,x3) = max{146, x1 + 81, x2 + 44, x3 + 82}
            if_active#!_A(x1,x2,x3) = max{x2 + 34, x3 + 1}
            false!_A = 187
            mark#!_A(x1) = max{4, x1 - 4}
            if!_A(x1,x2,x3) = max{191, x1 + 187, x2 + 35, x3 + 2}
            mark!_A(x1) = max{186, x1 + 129}
            true!_A = 187
            div!_A(x1,x2) = x1 + 196
            div_active#!_A(x1,x2) = max{84, x1 + 81}
            s!_A(x1) = 139
            ge_active!_A(x1,x2) = max{209, x1}
            minus!_A(x1,x2) = max{254, x1 - 1, x2 - 13}
            |0|!_A = 0
            minus_active!_A(x1,x2) = max{266, x1 + 255, x2 - 12}
            ge!_A(x1,x2) = 209
            div_active!_A(x1,x2) = x1 + 197
            if_active!_A(x1,x2,x3) = max{209, x1 + 188, x2 + 152, x3 + 205}
        
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(if) = [1, 2, 3]
      pi(mark) = [1]
      pi(true) = []
      pi(div) = [1]
      pi(div_active#) = []
      pi(s) = []
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(minus_active) = []
      pi(ge) = []
      pi(div_active) = [1]
      pi(if_active) = [1, 2, 3]

The following rules are removed:

  p1, p2, p3, p4, p5, p6, p7, p8

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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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 + 1
              minus_active#!_A(x1,x2) = x1 + x2 + 3
              s!_A(x1) = 2
        
        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}
            minus_active#!_A(x1,x2) = max{6, x1 - 10, x2 - 11}
            s!_A(x1) = max{5, x1 + 2}
        
  
    partial status:
  
      pi(minus_active#) = [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: 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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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 + 1
              ge_active#!_A(x1,x2) = 0
              s!_A(x1) = x1 + 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            ge_active#_A(x1,x2) = max{x1, x2}
            s_A(x1) = max{57, x1 + 51}
            ge_active#!_A(x1,x2) = max{4, x1 - 46}
            s!_A(x1) = max{27, x1 + 3}
        
  
    partial status:
  
      pi(ge_active#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.