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, r5, r7, r11, r15, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_active#_A(x1,x2,x3) = max{x2, x3 + 3}
        false_A = 1
        mark#_A(x1) = max{2, x1}
        if_A(x1,x2,x3) = max{x1 + 14, x2 + 14, x3 + 14}
        mark_A(x1) = max{16, x1 + 2}
        true_A = 1
        div_A(x1,x2) = max{x1 + 5, x2}
        div_active#_A(x1,x2) = max{5, x2}
        s_A(x1) = max{11, x1}
        ge_active_A(x1,x2) = max{30, x1 - 12, x2 + 19}
        minus_A(x1,x2) = max{6, x2 - 15}
        |0|_A = 8
        minus_active_A(x1,x2) = max{17, x1 - 12, x2 - 12}
        div_active_A(x1,x2) = max{3, x2 + 1}
        ge_A(x1,x2) = max{1, x1 - 15, x2 - 3}
  
    base order 2:
  
      precedence:
      
        mark = s = ge_active = minus = minus_active > if = true = div = ge > if_active# = mark# = div_active# = div_active > |0| > false
      
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(if) = [1]
      pi(mark) = [1]
      pi(true) = []
      pi(div) = []
      pi(div_active#) = []
      pi(s) = [1]
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(minus_active) = []
      pi(div_active) = []
      pi(ge) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: mark#(if(x,y,z)) -> mark#(x)
p2: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)
p3: if_active#(true(),x,y) -> mark#(x)
p4: mark#(div(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|())
p7: 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, p6, p7}


-- Reduction pair.

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

p1: mark#(if(x,y,z)) -> 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|())
p5: if_active#(true(),x,y) -> mark#(x)
p6: mark#(div(x,y)) -> mark#(x)
p7: mark#(if(x,y,z)) -> if_active#(mark(x),y,z)

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, r5, r7, r11, r15, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mark#_A(x1) = x1 + 3
        if_A(x1,x2,x3) = max{31, x1 + 5, x2 + 4, x3 + 28}
        s_A(x1) = max{8, x1}
        div_A(x1,x2) = max{x1 + 16, x2 + 17}
        div_active#_A(x1,x2) = x2 + 20
        mark_A(x1) = max{0, x1 - 31}
        if_active#_A(x1,x2,x3) = max{x2 + 3, x3 + 28}
        ge_active_A(x1,x2) = max{9, x1 - 31, x2 - 31}
        minus_A(x1,x2) = max{9, x2 - 18}
        |0|_A = 0
        true_A = 1
        minus_active_A(x1,x2) = max{10, x1 - 9, x2 - 18}
        false_A = 0
        div_active_A(x1,x2) = max{15, x1 - 1, x2}
        ge_A(x1,x2) = max{8, x1 - 32, x2 - 32}
  
    base order 2:
  
      precedence:
      
        div = minus_active > s = ge_active = div_active > if = minus = true = ge > mark# = div_active# = mark = if_active# > |0| > false
      
  
    partial status:
  
      pi(mark#) = []
      pi(if) = [2]
      pi(s) = []
      pi(div) = [2]
      pi(div_active#) = []
      pi(mark) = []
      pi(if_active#) = []
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(minus_active) = []
      pi(false) = []
      pi(div_active) = [2]
      pi(ge) = []

The following rules are removed:

  p7



-- SCC decomposition.

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

p1: mark#(if(x,y,z)) -> 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|())
p5: if_active#(true(),x,y) -> mark#(x)
p6: mark#(div(x,y)) -> 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, p6}


-- Reduction pair.

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

p1: mark#(if(x,y,z)) -> mark#(x)
p2: mark#(div(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: if_active#(true(),x,y) -> mark#(x)
p6: 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, r5, r6, r7, r8, r9, r15, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mark#_A(x1) = max{10, x1 - 6}
        if_A(x1,x2,x3) = max{19, x1 + 17, x2 - 32, x3 + 17}
        div_A(x1,x2) = max{x1 + 30, x2 + 14}
        div_active#_A(x1,x2) = max{15, x2 + 8}
        mark_A(x1) = max{18, x1}
        s_A(x1) = max{16, x1}
        if_active#_A(x1,x2,x3) = max{x2 - 6, x3 + 11}
        ge_active_A(x1,x2) = max{x1 - 20, x2 + 7}
        minus_A(x1,x2) = max{0, x2 - 16}
        |0|_A = 12
        true_A = 11
        minus_active_A(x1,x2) = max{18, x2 - 16}
        false_A = 11
        ge_A(x1,x2) = max{x1 - 20, x2 + 7}
  
    base order 2:
  
      precedence:
      
        |0| > if = true > mark > div > mark# = div_active# = if_active# > ge_active > s = ge > minus_active = false > minus
      
  
    partial status:
  
      pi(mark#) = []
      pi(if) = [1]
      pi(div) = []
      pi(div_active#) = []
      pi(mark) = [1]
      pi(s) = [1]
      pi(if_active#) = []
      pi(ge_active) = [2]
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(minus_active) = []
      pi(false) = []
      pi(ge) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: mark#(div(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: if_active#(true(),x,y) -> mark#(x)
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#(div(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|())
p5: if_active#(true(),x,y) -> 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, r5, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mark#_A(x1) = max{5, x1 - 1}
        div_A(x1,x2) = max{25, x1 + 7, x2 + 13}
        s_A(x1) = max{23, x1}
        div_active#_A(x1,x2) = x2 + 12
        mark_A(x1) = 32
        if_active#_A(x1,x2,x3) = max{35, x2 - 1, x3 + 4}
        ge_active_A(x1,x2) = max{x1 + 6, x2 + 13}
        minus_A(x1,x2) = max{27, x2 + 6}
        |0|_A = 31
        true_A = 4
        minus_active_A(x1,x2) = max{33, x1 - 32}
        if_active_A(x1,x2,x3) = max{x1 - 14, x3 + 33}
        if_A(x1,x2,x3) = x3 + 26
  
    base order 2:
  
      precedence:
      
        minus_active = if_active > mark > div = true = if > mark# = div_active# = if_active# > minus > s > |0| > ge_active
      
  
    partial status:
  
      pi(mark#) = []
      pi(div) = []
      pi(s) = []
      pi(div_active#) = []
      pi(mark) = []
      pi(if_active#) = []
      pi(ge_active) = [2]
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(minus_active) = []
      pi(if_active) = []
      pi(if) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: mark#(s(x)) -> 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: if_active#(true(),x,y) -> 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: mark#(s(x)) -> 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: if_active#(true(),x,y) -> 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, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mark#_A(x1) = max{14, x1 + 3}
        s_A(x1) = max{63, x1 + 12}
        div_A(x1,x2) = max{51, x1 + 10}
        div_active#_A(x1,x2) = max{50, x1 + 4}
        mark_A(x1) = max{8, x1}
        if_active#_A(x1,x2,x3) = max{x2 + 4, x3 + 15}
        ge_active_A(x1,x2) = max{x1 + 18, x2 + 1}
        minus_A(x1,x2) = 23
        |0|_A = 17
        true_A = 5
        minus_active_A(x1,x2) = 23
        false_A = 7
        ge_A(x1,x2) = max{x1 + 18, x2 + 1}
        div_active_A(x1,x2) = max{51, x1 + 10}
        if_A(x1,x2,x3) = max{73, x2 + 7, x3 + 1}
        if_active_A(x1,x2,x3) = max{73, x2 + 7, x3 + 1}
  
    base order 2:
  
      precedence:
      
        mark > div_active > s = minus = minus_active = if_active > ge_active = ge = if > true > mark# = |0| > div = div_active# = false > if_active#
      
  
    partial status:
  
      pi(mark#) = []
      pi(s) = [1]
      pi(div) = []
      pi(div_active#) = []
      pi(mark) = [1]
      pi(if_active#) = []
      pi(ge_active) = [2]
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(minus_active) = []
      pi(false) = []
      pi(ge) = [2]
      pi(div_active) = [1]
      pi(if) = [3]
      pi(if_active) = []

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: mark#(s(x)) -> 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|())

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

  r1, r2, r5, r7, r9, r11, r17, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mark#_A(x1) = x1 + 11
        s_A(x1) = x1 + 11
        minus_active_A(x1,x2) = max{3, x1 - 63, x2 - 5}
        |0|_A = 2
        mark_A(x1) = max{5, x1 - 41}
        ge_active_A(x1,x2) = max{12, x1 - 41, x2 - 41}
        true_A = 1
        false_A = 1
        div_active_A(x1,x2) = max{41, x1 - 31, x2 - 13}
        ge_A(x1,x2) = max{11, x1 - 42, x2 - 42}
        div_A(x1,x2) = max{29, x1 - 40, x2 - 25}
  
    base order 2:
  
      precedence:
      
        div_active > ge_active = div > minus_active = ge > s = |0| > mark# = mark > true = false
      
  
    partial status:
  
      pi(mark#) = [1]
      pi(s) = [1]
      pi(minus_active) = []
      pi(|0|) = []
      pi(mark) = []
      pi(ge_active) = []
      pi(true) = []
      pi(false) = []
      pi(div_active) = []
      pi(ge) = []
      pi(div) = []

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

  r1, r2, r5, r7, r11, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_active#_A(x1,x2) = max{2, x1 + 1}
        s_A(x1) = x1 + 7
        minus_active_A(x1,x2) = max{5, x2 - 36}
        |0|_A = 4
        mark_A(x1) = max{6, x1 - 34}
        ge_active_A(x1,x2) = max{8, x1 - 26, x2 - 36}
        true_A = 1
        false_A = 2
        div_active_A(x1,x2) = max{10, x1 - 32, x2 - 42}
        ge_A(x1,x2) = max{7, x1 - 27, x2 - 37}
        if_active_A(x1,x2,x3) = max{8, x1 - 9, x2 - 23, x3 - 5}
        if_A(x1,x2,x3) = max{6, x1 - 34, x2 - 24, x3 - 6}
  
    base order 2:
  
      precedence:
      
        if_active > minus_active# = s > div_active > ge_active = if > mark = ge > minus_active > |0| > true = false
      
  
    partial status:
  
      pi(minus_active#) = [1]
      pi(s) = [1]
      pi(minus_active) = []
      pi(|0|) = []
      pi(mark) = []
      pi(ge_active) = []
      pi(true) = []
      pi(false) = []
      pi(div_active) = []
      pi(ge) = []
      pi(if_active) = []
      pi(if) = []

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, r5, r7, r11, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ge_active#_A(x1,x2) = max{3, x1 + 2}
        s_A(x1) = x1 + 7
        minus_active_A(x1,x2) = max{5, x2 - 30}
        |0|_A = 4
        mark_A(x1) = max{6, x1 - 28}
        ge_active_A(x1,x2) = max{8, x1 - 19, x2 - 36}
        true_A = 1
        false_A = 3
        div_active_A(x1,x2) = max{9, x1 - 25, x2 - 42}
        ge_A(x1,x2) = max{7, x1 - 20, x2 - 37}
        if_active_A(x1,x2,x3) = max{8, x1 - 9, x2 - 16, x3 - 5}
        if_A(x1,x2,x3) = max{6, x1 - 28, x2 - 17, x3 - 6}
        div_A(x1,x2) = max{8, x1 - 26, x2 - 43}
  
    base order 2:
  
      precedence:
      
        if_active = div > ge_active# = s > div_active > ge_active = if > mark = ge > minus_active > |0| > true = false
      
  
    partial status:
  
      pi(ge_active#) = [1]
      pi(s) = [1]
      pi(minus_active) = []
      pi(|0|) = []
      pi(mark) = []
      pi(ge_active) = []
      pi(true) = []
      pi(false) = []
      pi(div_active) = []
      pi(ge) = []
      pi(if_active) = []
      pi(if) = []
      pi(div) = []

The following rules are removed:

  p1

Then no dependency pair remains.