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

  (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!):
        
          max/plus interpretation on natural numbers:
          
            if_active#_A(x1,x2,x3) = max{61, x2 - 17, x3 - 17}
            false_A = 0
            mark#_A(x1) = max{61, x1 - 17}
            if_A(x1,x2,x3) = max{x1 + 1, x2, x3 + 77}
            mark_A(x1) = max{18, x1}
            true_A = 19
            div_A(x1,x2) = max{x1 + 12, x2 + 49}
            div_active#_A(x1,x2) = max{61, x2 + 32}
            s_A(x1) = max{4, x1}
            ge_active_A(x1,x2) = 0
            minus_A(x1,x2) = 6
            |0|_A = 49
            if_active#!_A(x1,x2,x3) = max{30, x2 + 24, x3 + 26}
            false!_A = 16
            mark#!_A(x1) = max{25, x1 + 24}
            if!_A(x1,x2,x3) = x3 + 29
            mark!_A(x1) = max{72, x1 - 4}
            true!_A = 1
            div!_A(x1,x2) = max{x1 + 7, x2 + 74}
            div_active#!_A(x1,x2) = x2 + 73
            s!_A(x1) = x1 + 79
            ge_active!_A(x1,x2) = 15
            minus!_A(x1,x2) = max{95, x1 - 1, x2 + 80}
            |0|!_A = 95
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_active#_A(x1,x2,x3) = 1
            false_A = 37
            mark#_A(x1) = 1
            if_A(x1,x2,x3) = max{55, x1 + 3, x2 + 41, x3 + 40}
            mark_A(x1) = max{48, x1 + 13}
            true_A = 48
            div_A(x1,x2) = max{16, x1 - 13}
            div_active#_A(x1,x2) = 1
            s_A(x1) = 2
            ge_active_A(x1,x2) = 48
            minus_A(x1,x2) = 28
            |0|_A = 8
            if_active#!_A(x1,x2,x3) = 54
            false!_A = 48
            mark#!_A(x1) = 54
            if!_A(x1,x2,x3) = max{54, x2 + 50, x3 + 50}
            mark!_A(x1) = 49
            true!_A = 55
            div!_A(x1,x2) = x1 + 59
            div_active#!_A(x1,x2) = 54
            s!_A(x1) = 47
            ge_active!_A(x1,x2) = max{57, x2 + 14}
            minus!_A(x1,x2) = max{57, x1 - 10}
            |0|!_A = 95
        
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(if) = []
      pi(mark) = []
      pi(true) = []
      pi(div) = []
      pi(div_active#) = []
      pi(s) = []
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []

The following rules are removed:

  p1, p2, p5



-- SCC decomposition.

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

p1: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)
p2: if_active#(true(),x,y) -> 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|())
p5: 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, p5}


-- Reduction pair.

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

p1: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)
p2: if_active#(true(),x,y) -> mark#(x)
p3: mark#(s(x)) -> mark#(x)
p4: mark#(div(x,y)) -> div_active#(mark(x),y)
p5: 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

  (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!):
        
          max/plus interpretation on natural numbers:
          
            mark#_A(x1) = 1
            if_A(x1,x2,x3) = max{165, x2 + 78}
            if_active#_A(x1,x2,x3) = 1
            mark_A(x1) = x1 + 85
            true_A = 4
            s_A(x1) = max{18, x1}
            div_A(x1,x2) = max{18, x2 + 17}
            div_active#_A(x1,x2) = 1
            ge_active_A(x1,x2) = 2
            minus_A(x1,x2) = max{165, x2 - 11}
            |0|_A = 76
            mark#!_A(x1) = x1 + 2
            if!_A(x1,x2,x3) = max{x2 + 40, x3 + 40}
            if_active#!_A(x1,x2,x3) = max{4, x2 + 2}
            mark!_A(x1) = 0
            true!_A = 2
            s!_A(x1) = x1 + 38
            div!_A(x1,x2) = max{17, x1 + 12}
            div_active#!_A(x1,x2) = x2 + 19
            ge_active!_A(x1,x2) = 38
            minus!_A(x1,x2) = 2
            |0|!_A = 4
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            mark#_A(x1) = 1
            if_A(x1,x2,x3) = max{0, x2 - 8, x3 - 7}
            if_active#_A(x1,x2,x3) = 1
            mark_A(x1) = 8
            true_A = 23
            s_A(x1) = max{9, x1}
            div_A(x1,x2) = max{14, x2 + 5}
            div_active#_A(x1,x2) = 1
            ge_active_A(x1,x2) = 9
            minus_A(x1,x2) = 0
            |0|_A = 19
            mark#!_A(x1) = x1 + 7
            if!_A(x1,x2,x3) = max{x1 + 31, x2 - 1, x3 - 1}
            if_active#!_A(x1,x2,x3) = max{x2 + 7, x3 + 1}
            mark!_A(x1) = 4
            true!_A = 0
            s!_A(x1) = x1 + 7
            div!_A(x1,x2) = x1 + 31
            div_active#!_A(x1,x2) = x2 + 12
            ge_active!_A(x1,x2) = 0
            minus!_A(x1,x2) = 0
            |0|!_A = 9
        
  
    partial status:
  
      pi(mark#) = []
      pi(if) = [1, 3]
      pi(if_active#) = []
      pi(mark) = []
      pi(true) = []
      pi(s) = []
      pi(div) = []
      pi(div_active#) = []
      pi(ge_active) = []
      pi(minus) = [2]
      pi(|0|) = []

The following rules are removed:

  p1



-- SCC decomposition.

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 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#(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 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!):
        
          max/plus interpretation on natural numbers:
          
            if_active#_A(x1,x2,x3) = max{60, x2 + 58, x3 - 1}
            true_A = 1
            mark#_A(x1) = 60
            div_A(x1,x2) = 28
            div_active#_A(x1,x2) = 60
            mark_A(x1) = 29
            s_A(x1) = 2
            ge_active_A(x1,x2) = 4
            minus_A(x1,x2) = 29
            |0|_A = 0
            minus_active_A(x1,x2) = 29
            false_A = 4
            ge_A(x1,x2) = 4
            div_active_A(x1,x2) = 29
            if_A(x1,x2,x3) = 4
            if_active_A(x1,x2,x3) = 29
            if_active#!_A(x1,x2,x3) = max{x1 + 30, x3 + 35}
            true!_A = 1
            mark#!_A(x1) = 35
            div!_A(x1,x2) = 3
            div_active#!_A(x1,x2) = 35
            mark!_A(x1) = 34
            s!_A(x1) = 2
            ge_active!_A(x1,x2) = 5
            minus!_A(x1,x2) = 0
            |0|!_A = 0
            minus_active!_A(x1,x2) = 0
            false!_A = 1
            ge!_A(x1,x2) = 1
            div_active!_A(x1,x2) = max{34, x1 + 4}
            if!_A(x1,x2,x3) = 1
            if_active!_A(x1,x2,x3) = 34
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_active#_A(x1,x2,x3) = max{1, x3 - 4}
            true_A = 66
            mark#_A(x1) = 1
            div_A(x1,x2) = x1 + 34
            div_active#_A(x1,x2) = 1
            mark_A(x1) = x1 + 10
            s_A(x1) = max{76, x1 + 22}
            ge_active_A(x1,x2) = x1 + 66
            minus_A(x1,x2) = max{29, x1 - 125}
            |0|_A = 3
            minus_active_A(x1,x2) = max{29, x1 - 115}
            false_A = 48
            ge_A(x1,x2) = x1 + 56
            div_active_A(x1,x2) = x1 + 34
            if_A(x1,x2,x3) = max{26, x1 - 10, x2 + 25, x3 + 22}
            if_active_A(x1,x2,x3) = max{36, x1 - 10, x2 + 25, x3 + 25}
            if_active#!_A(x1,x2,x3) = max{x2 + 68, x3 + 96}
            true!_A = 87
            mark#!_A(x1) = max{88, x1 + 67}
            div!_A(x1,x2) = 0
            div_active#!_A(x1,x2) = x1 + 78
            mark!_A(x1) = max{4, x1 - 22}
            s!_A(x1) = 78
            ge_active!_A(x1,x2) = 3
            minus!_A(x1,x2) = max{0, x1 - 149}
            |0|!_A = 5
            minus_active!_A(x1,x2) = max{6, x1 - 148}
            false!_A = 2
            ge!_A(x1,x2) = 2
            div_active!_A(x1,x2) = max{2, x1 + 1}
            if!_A(x1,x2,x3) = max{1, x3 - 21}
            if_active!_A(x1,x2,x3) = max{1, x1 - 43, x2 - 8, x3 - 21}
        
  
    partial status:
  
      pi(if_active#) = []
      pi(true) = []
      pi(mark#) = []
      pi(div) = []
      pi(div_active#) = [1]
      pi(mark) = []
      pi(s) = []
      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

  (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!):
        
          max/plus interpretation on natural numbers:
          
            minus_active#_A(x1,x2) = 0
            s_A(x1) = max{9, x1 - 64}
            minus_active#!_A(x1,x2) = max{1, x2 - 51}
            s!_A(x1) = max{4, x1 - 50}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus_active#_A(x1,x2) = 0
            s_A(x1) = max{9, x1 - 48}
            minus_active#!_A(x1,x2) = max{24, x1 - 10, x2 - 10}
            s!_A(x1) = 23
        
  
    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:
  
      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!):
        
          max/plus interpretation on natural numbers:
          
            ge_active#_A(x1,x2) = 0
            s_A(x1) = max{5, x1 - 17}
            ge_active#!_A(x1,x2) = max{0, x2 - 56}
            s!_A(x1) = max{7, x1 - 55}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            ge_active#_A(x1,x2) = 0
            s_A(x1) = max{1, x1 - 6}
            ge_active#!_A(x1,x2) = max{1, x1 - 2, x2 - 2}
            s!_A(x1) = 0
        
  
    partial status:
  
      pi(ge_active#) = [2]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.