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, >MG)
  
    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 + 2
          false_A() = 0
          mark#_A(x1) = x1 + 2
          if_A(x1,x2,x3) = x1 + x2 + x3
          mark_A(x1) = x1
          true_A() = 0
          div_A(x1,x2) = x1 + x2 + 3
          div_active#_A(x1,x2) = x1 + x2 + 5
          s_A(x1) = x1 + 1
          ge_active_A(x1,x2) = 0
          minus_A(x1,x2) = 0
          |0|_A() = 0
          minus_active_A(x1,x2) = 0
          ge_A(x1,x2) = 0
          div_active_A(x1,x2) = x1 + x2 + 3
          if_active_A(x1,x2,x3) = x1 + x2 + x3
  
    base order 2:
  
      precedence:
      
        if_active# = mark# = div_active# > if = mark = minus = minus_active = div_active = if_active > div > s > true = ge_active > false = ge > |0|
      
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(if) = []
      pi(mark) = []
      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) = []
      pi(if_active) = []

The following rules are removed:

  p5



-- SCC decomposition.

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)) -> 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: if_active#(false(),x,y) -> mark#(y)
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#(if(x,y,z)) -> if_active#(mark(x),y,z)
p7: mark#(if(x,y,z)) -> 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, r7, r11, r15, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    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 + 2
          false_A() = 1
          mark#_A(x1) = x1 + 2
          s_A(x1) = x1
          div_A(x1,x2) = 3
          div_active#_A(x1,x2) = 5
          mark_A(x1) = x1 + 2
          ge_active_A(x1,x2) = x2 + 2
          minus_A(x1,x2) = x2 + 6
          |0|_A() = 0
          true_A() = 1
          if_A(x1,x2,x3) = x1 + x2 + x3 + 5
          minus_active_A(x1,x2) = x1 + x2 + 7
          div_active_A(x1,x2) = 1
          ge_A(x1,x2) = 1
  
    base order 2:
  
      precedence:
      
        minus_active > false = minus = if > s = mark = ge_active = ge > if_active# = mark# = div_active# > div > div_active > |0| > true
      
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(s) = [1]
      pi(div) = []
      pi(div_active#) = []
      pi(mark) = [1]
      pi(ge_active) = [2]
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(if) = []
      pi(minus_active) = [1, 2]
      pi(div_active) = []
      pi(ge) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

p1: if_active#(false(),x,y) -> mark#(y)
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#(if(x,y,z)) -> 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: if_active#(false(),x,y) -> mark#(y)
p2: mark#(if(x,y,z)) -> 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, r6, r11, r15, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    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 + 1
          false_A() = 4
          mark#_A(x1) = x1 + 1
          if_A(x1,x2,x3) = x1 + 2
          div_A(x1,x2) = x1 + 1
          div_active#_A(x1,x2) = 2
          mark_A(x1) = 0
          s_A(x1) = x1
          ge_active_A(x1,x2) = 3
          minus_A(x1,x2) = 0
          |0|_A() = 0
          true_A() = 4
          minus_active_A(x1,x2) = 0
          div_active_A(x1,x2) = x2
          ge_A(x1,x2) = 1
  
    base order 2:
  
      precedence:
      
        if > if_active# = mark# = div_active# > s > false = div = ge_active > mark = ge > minus_active > minus = div_active > |0| > true
      
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(if) = []
      pi(div) = []
      pi(div_active#) = []
      pi(mark) = []
      pi(s) = []
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(minus_active) = []
      pi(div_active) = [2]
      pi(ge) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: if_active#(false(),x,y) -> mark#(y)
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: if_active#(false(),x,y) -> mark#(y)
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, 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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_active#_A(x1,x2,x3) = x2 + x3
          false_A() = 1
          mark#_A(x1) = x1
          s_A(x1) = x1 + 3
          div_A(x1,x2) = x1 + x2 + 1
          div_active#_A(x1,x2) = x1 + x2 + 1
          mark_A(x1) = x1
          ge_active_A(x1,x2) = 2
          minus_A(x1,x2) = x1
          |0|_A() = 0
          true_A() = 1
          minus_active_A(x1,x2) = x1
          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:
  
      precedence:
      
        div = div_active > minus = minus_active > div_active# = if = if_active > s = ge_active = ge > if_active# > mark# > true > mark > |0| > false
      
  
    partial status:
  
      pi(if_active#) = [2, 3]
      pi(false) = []
      pi(mark#) = [1]
      pi(s) = []
      pi(div) = [2]
      pi(div_active#) = []
      pi(mark) = [1]
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(minus_active) = []
      pi(ge) = []
      pi(div_active) = [2]
      pi(if) = [2, 3]
      pi(if_active) = [2, 3]

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: if_active#(false(),x,y) -> mark#(y)
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#(false(),x,y) -> mark#(y)
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, r5, r7, r9, r11, r15, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_active#_A(x1,x2,x3) = x3 + 5
          false_A() = 3
          mark#_A(x1) = x1 + 4
          div_A(x1,x2) = x2 + 13
          div_active#_A(x1,x2) = 12
          mark_A(x1) = 2
          s_A(x1) = x1 + 7
          ge_active_A(x1,x2) = x1 + x2 + 11
          minus_A(x1,x2) = 9
          |0|_A() = 1
          minus_active_A(x1,x2) = x1 + x2 + 10
          true_A() = 3
          div_active_A(x1,x2) = x2 + 3
          ge_A(x1,x2) = 1
          if_active_A(x1,x2,x3) = x2 + 3
          if_A(x1,x2,x3) = 1
  
    base order 2:
  
      precedence:
      
        if_active# > mark# = div = minus_active > s > false = true > mark > ge_active = if_active > div_active = ge = if > minus > div_active# > |0|
      
  
    partial status:
  
      pi(if_active#) = []
      pi(false) = []
      pi(mark#) = []
      pi(div) = []
      pi(div_active#) = []
      pi(mark) = []
      pi(s) = []
      pi(ge_active) = []
      pi(minus) = []
      pi(|0|) = []
      pi(minus_active) = [2]
      pi(true) = []
      pi(div_active) = []
      pi(ge) = []
      pi(if_active) = [2]
      pi(if) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: mark#(div(x,y)) -> div_active#(mark(x),y)
p2: div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),|0|())
p3: 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:

  {p3}


-- 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, r3, r5, r7, r9, r11, r15, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          mark#_A(x1) = x1
          s_A(x1) = x1 + 6
          minus_active_A(x1,x2) = x1 + x2 + 7
          |0|_A() = 3
          mark_A(x1) = 4
          ge_active_A(x1,x2) = x2 + 3
          true_A() = 5
          false_A() = 5
          div_active_A(x1,x2) = x2 + 7
          minus_A(x1,x2) = 1
          ge_A(x1,x2) = 1
          if_active_A(x1,x2,x3) = x2 + 6
          if_A(x1,x2,x3) = 5
          div_A(x1,x2) = 0
  
    base order 2:
  
      precedence:
      
        mark# > s = minus_active = div_active = div > true = false = minus > |0| = if > mark = if_active > ge_active = ge
      
  
    partial status:
  
      pi(mark#) = []
      pi(s) = []
      pi(minus_active) = []
      pi(|0|) = []
      pi(mark) = []
      pi(ge_active) = []
      pi(true) = []
      pi(false) = []
      pi(div_active) = [2]
      pi(minus) = []
      pi(ge) = []
      pi(if_active) = [2]
      pi(if) = []
      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, r3, r5, r7, r11, r15, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    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 + 5
          minus_active_A(x1,x2) = x2 + 4
          |0|_A() = 3
          mark_A(x1) = 8
          ge_active_A(x1,x2) = x1 + x2 + 7
          true_A() = 9
          false_A() = 9
          div_active_A(x1,x2) = x2 + 1
          minus_A(x1,x2) = 1
          ge_A(x1,x2) = 1
          if_active_A(x1,x2,x3) = x1 + x2 + 10
          if_A(x1,x2,x3) = 9
          div_A(x1,x2) = 0
  
    base order 2:
  
      precedence:
      
        minus_active# = ge_active = if_active > true = false = ge = if > mark = div_active = div > s > minus > minus_active > |0|
      
  
    partial status:
  
      pi(minus_active#) = [2]
      pi(s) = []
      pi(minus_active) = [2]
      pi(|0|) = []
      pi(mark) = []
      pi(ge_active) = [1, 2]
      pi(true) = []
      pi(false) = []
      pi(div_active) = [2]
      pi(minus) = []
      pi(ge) = []
      pi(if_active) = [2]
      pi(if) = []
      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: 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, r3, r5, r7, r11, r15, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    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 + 5
          minus_active_A(x1,x2) = x2 + 4
          |0|_A() = 3
          mark_A(x1) = 8
          ge_active_A(x1,x2) = x1 + x2 + 7
          true_A() = 9
          false_A() = 9
          div_active_A(x1,x2) = x2 + 1
          minus_A(x1,x2) = 1
          ge_A(x1,x2) = 1
          if_active_A(x1,x2,x3) = x1 + x2 + 10
          if_A(x1,x2,x3) = 9
          div_A(x1,x2) = 0
  
    base order 2:
  
      precedence:
      
        ge_active# = ge_active = if_active > true = false = ge = if > mark = div_active = div > s > minus > minus_active > |0|
      
  
    partial status:
  
      pi(ge_active#) = [2]
      pi(s) = []
      pi(minus_active) = [2]
      pi(|0|) = []
      pi(mark) = []
      pi(ge_active) = [1, 2]
      pi(true) = []
      pi(false) = []
      pi(div_active) = [2]
      pi(minus) = []
      pi(ge) = []
      pi(if_active) = [2]
      pi(if) = []
      pi(div) = []

The following rules are removed:

  p1

Then no dependency pair remains.