YES

We show the termination of the TRS R:

  a__minus(|0|(),Y) -> |0|()
  a__minus(s(X),s(Y)) -> a__minus(X,Y)
  a__geq(X,|0|()) -> true()
  a__geq(|0|(),s(Y)) -> false()
  a__geq(s(X),s(Y)) -> a__geq(X,Y)
  a__div(|0|(),s(Y)) -> |0|()
  a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
  a__if(true(),X,Y) -> mark(X)
  a__if(false(),X,Y) -> mark(Y)
  mark(minus(X1,X2)) -> a__minus(X1,X2)
  mark(geq(X1,X2)) -> a__geq(X1,X2)
  mark(div(X1,X2)) -> a__div(mark(X1),X2)
  mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
  mark(|0|()) -> |0|()
  mark(s(X)) -> s(mark(X))
  mark(true()) -> true()
  mark(false()) -> false()
  a__minus(X1,X2) -> minus(X1,X2)
  a__geq(X1,X2) -> geq(X1,X2)
  a__div(X1,X2) -> div(X1,X2)
  a__if(X1,X2,X3) -> if(X1,X2,X3)

-- SCC decomposition.

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

p1: a__minus#(s(X),s(Y)) -> a__minus#(X,Y)
p2: a__geq#(s(X),s(Y)) -> a__geq#(X,Y)
p3: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p4: a__div#(s(X),s(Y)) -> a__geq#(X,Y)
p5: a__if#(true(),X,Y) -> mark#(X)
p6: a__if#(false(),X,Y) -> mark#(Y)
p7: mark#(minus(X1,X2)) -> a__minus#(X1,X2)
p8: mark#(geq(X1,X2)) -> a__geq#(X1,X2)
p9: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p10: mark#(div(X1,X2)) -> mark#(X1)
p11: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
p12: mark#(if(X1,X2,X3)) -> mark#(X1)
p13: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

The estimated dependency graph contains the following SCCs:

  {p3, p5, p6, p9, p10, p11, p12, p13}
  {p1}
  {p2}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(if(X1,X2,X3)) -> mark#(X1)
p3: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
p4: a__if#(false(),X,Y) -> mark#(Y)
p5: mark#(div(X1,X2)) -> mark#(X1)
p6: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p7: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p8: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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, r20, r21

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{13, x1}
        s_A(x1) = max{78, x1 + 20}
        if_A(x1,x2,x3) = max{x1 + 12, x2 + 34, x3 + 157}
        a__if#_A(x1,x2,x3) = max{x1, x2 + 34, x3 + 156}
        mark_A(x1) = max{21, x1 + 12}
        false_A = 14
        div_A(x1,x2) = max{117, x1 + 107}
        a__div#_A(x1,x2) = x1 + 93
        a__geq_A(x1,x2) = x1 + 112
        minus_A(x1,x2) = 5
        |0|_A = 15
        true_A = 14
        a__minus_A(x1,x2) = 16
        a__div_A(x1,x2) = max{118, x1 + 107}
        a__if_A(x1,x2,x3) = max{x1 + 12, x2 + 46, x3 + 169}
        geq_A(x1,x2) = x1 + 104
  
    precedence:
    
      a__div# = a__geq = minus = true = geq > mark# = false = |0| > mark = div = a__div = a__if > a__minus > if > s = a__if#
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = []
      pi(if) = []
      pi(a__if#) = [2, 3]
      pi(mark) = []
      pi(false) = []
      pi(div) = []
      pi(a__div#) = [1]
      pi(a__geq) = [1]
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(a__minus) = []
      pi(a__div) = []
      pi(a__if) = []
      pi(geq) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(if(X1,X2,X3)) -> mark#(X1)
p3: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
p4: mark#(div(X1,X2)) -> mark#(X1)
p5: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p6: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p7: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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#(s(X)) -> mark#(X)
p2: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p3: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p4: a__if#(true(),X,Y) -> mark#(X)
p5: mark#(div(X1,X2)) -> mark#(X1)
p6: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
p7: mark#(if(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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, r20, r21

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 37
        s_A(x1) = max{18, x1}
        div_A(x1,x2) = max{x1 + 35, x2 + 37}
        a__div#_A(x1,x2) = max{x1 + 56, x2 + 74}
        mark_A(x1) = x1 + 15
        a__if#_A(x1,x2,x3) = max{x1 + 2, x2 + 37, x3 + 46}
        a__geq_A(x1,x2) = x1 + 30
        minus_A(x1,x2) = max{1, x1 - 16}
        |0|_A = 4
        true_A = 29
        if_A(x1,x2,x3) = max{x1 + 4, x2 + 5, x3 + 10}
        a__minus_A(x1,x2) = max{6, x1 - 12}
        a__div_A(x1,x2) = max{x1 + 35, x2 + 52}
        a__if_A(x1,x2,x3) = max{x1 + 4, x2 + 15, x3 + 25}
        false_A = 0
        geq_A(x1,x2) = x1 + 29
  
    precedence:
    
      s = mark = a__geq = minus = a__div = a__if > div = geq > mark# = a__div# = a__if# = |0| = true = a__minus = false > if
    
    partial status:
  
      pi(mark#) = []
      pi(s) = []
      pi(div) = [2]
      pi(a__div#) = []
      pi(mark) = []
      pi(a__if#) = []
      pi(a__geq) = []
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(if) = [1, 3]
      pi(a__minus) = []
      pi(a__div) = []
      pi(a__if) = []
      pi(false) = []
      pi(geq) = [1]

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p3: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p4: a__if#(true(),X,Y) -> mark#(X)
p5: mark#(div(X1,X2)) -> mark#(X1)
p6: mark#(if(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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#(s(X)) -> mark#(X)
p2: mark#(if(X1,X2,X3)) -> mark#(X1)
p3: mark#(div(X1,X2)) -> mark#(X1)
p4: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p5: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p6: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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, r20, r21

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 15
        s_A(x1) = max{11, x1}
        if_A(x1,x2,x3) = max{x1 + 1, x2, x3 + 2}
        div_A(x1,x2) = max{x1 + 14, x2 + 12}
        a__div#_A(x1,x2) = max{x1 + 6, x2 + 27}
        mark_A(x1) = x1 + 18
        a__if#_A(x1,x2,x3) = max{x1 - 2, x2 + 15, x3 + 11}
        a__geq_A(x1,x2) = x2 + 29
        minus_A(x1,x2) = 2
        |0|_A = 1
        true_A = 13
        a__minus_A(x1,x2) = 2
        a__div_A(x1,x2) = max{x1 + 14, x2 + 30}
        a__if_A(x1,x2,x3) = max{x1 + 1, x2 + 18, x3 + 19}
        false_A = 0
        geq_A(x1,x2) = x2 + 29
  
    precedence:
    
      a__geq = geq > mark# = s = if = div = a__div# = mark = a__if# = minus = |0| = true = a__minus = a__div = a__if = false
    
    partial status:
  
      pi(mark#) = []
      pi(s) = []
      pi(if) = []
      pi(div) = []
      pi(a__div#) = []
      pi(mark) = []
      pi(a__if#) = []
      pi(a__geq) = []
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(a__minus) = []
      pi(a__div) = []
      pi(a__if) = []
      pi(false) = []
      pi(geq) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(div(X1,X2)) -> mark#(X1)
p3: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p4: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p5: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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#(s(X)) -> mark#(X)
p2: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p3: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p4: a__if#(true(),X,Y) -> mark#(X)
p5: mark#(div(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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, r20, r21

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{0, x1 - 38}
        s_A(x1) = max{15, x1}
        div_A(x1,x2) = max{x1 + 118, x2 + 167}
        a__div#_A(x1,x2) = x2 + 129
        mark_A(x1) = x1 + 67
        a__if#_A(x1,x2,x3) = max{128, x1 + 14, x2 - 38, x3 + 114}
        a__geq_A(x1,x2) = 32
        minus_A(x1,x2) = max{27, x2 + 11}
        |0|_A = 30
        true_A = 31
        a__minus_A(x1,x2) = max{47, x2 + 32}
        a__div_A(x1,x2) = max{x1 + 118, x2 + 234}
        a__if_A(x1,x2,x3) = max{x2 + 67, x3 + 90}
        false_A = 14
        if_A(x1,x2,x3) = max{x2 + 48, x3 + 23}
        geq_A(x1,x2) = 31
  
    precedence:
    
      minus > a__geq = true = geq > |0| > false > a__minus > mark# = s = div = a__div# = mark = a__if# = a__div = a__if = if
    
    partial status:
  
      pi(mark#) = []
      pi(s) = []
      pi(div) = []
      pi(a__div#) = []
      pi(mark) = []
      pi(a__if#) = []
      pi(a__geq) = []
      pi(minus) = [2]
      pi(|0|) = []
      pi(true) = []
      pi(a__minus) = []
      pi(a__div) = []
      pi(a__if) = []
      pi(false) = []
      pi(if) = []
      pi(geq) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p3: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p4: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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(X1,X2)) -> a__div#(mark(X1),X2)
p3: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p4: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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, r20, r21

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 9
        s_A(x1) = x1 + 16
        div_A(x1,x2) = x1 + 11
        a__div#_A(x1,x2) = x1 + 20
        mark_A(x1) = x1
        a__if#_A(x1,x2,x3) = max{x1 - 5, x2 + 9, x3 + 36}
        a__geq_A(x1,x2) = x1 + 39
        minus_A(x1,x2) = max{0, x1 - 17}
        |0|_A = 0
        true_A = 38
        a__minus_A(x1,x2) = max{0, x1 - 17}
        a__div_A(x1,x2) = x1 + 11
        a__if_A(x1,x2,x3) = max{x2, x3}
        false_A = 15
        if_A(x1,x2,x3) = max{x2, x3}
        geq_A(x1,x2) = x1 + 39
  
    precedence:
    
      mark = a__div = a__if = if > mark# = div = a__div# = a__if# = minus = |0| = a__minus = false > s = a__geq = true = geq
    
    partial status:
  
      pi(mark#) = []
      pi(s) = [1]
      pi(div) = []
      pi(a__div#) = []
      pi(mark) = []
      pi(a__if#) = []
      pi(a__geq) = []
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(a__minus) = []
      pi(a__div) = []
      pi(a__if) = []
      pi(false) = []
      pi(if) = []
      pi(geq) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p2: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p3: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p2: a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
p3: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

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, r20, r21

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{1, x1}
        div_A(x1,x2) = x2 + 5
        a__div#_A(x1,x2) = x2 + 5
        mark_A(x1) = max{19, x1 + 12}
        s_A(x1) = x1
        a__if#_A(x1,x2,x3) = max{x1 - 12, x2, x3 + 2}
        a__geq_A(x1,x2) = 17
        minus_A(x1,x2) = max{x1 + 2, x2 + 2}
        |0|_A = 0
        true_A = 16
        a__minus_A(x1,x2) = max{x1 + 2, x2 + 2}
        a__div_A(x1,x2) = max{19, x2 + 17}
        a__if_A(x1,x2,x3) = max{x1 - 8, x2 + 12, x3 + 19}
        false_A = 0
        if_A(x1,x2,x3) = max{x1 - 8, x2 + 12, x3 + 19}
        geq_A(x1,x2) = 17
  
    precedence:
    
      a__geq = minus = |0| = true = a__minus = false > mark = a__div = a__if > div = a__div# = geq > a__if# = if > mark# = s
    
    partial status:
  
      pi(mark#) = [1]
      pi(div) = [2]
      pi(a__div#) = []
      pi(mark) = []
      pi(s) = []
      pi(a__if#) = [2]
      pi(a__geq) = []
      pi(minus) = []
      pi(|0|) = []
      pi(true) = []
      pi(a__minus) = []
      pi(a__div) = []
      pi(a__if) = []
      pi(false) = []
      pi(if) = []
      pi(geq) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
p2: a__if#(true(),X,Y) -> mark#(X)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: a__minus#(s(X),s(Y)) -> a__minus#(X,Y)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        a__minus#_A(x1,x2) = max{0, x1 - 2, x2 - 2}
        s_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      a__minus# = s
    
    partial status:
  
      pi(a__minus#) = []
      pi(s) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: a__geq#(s(X),s(Y)) -> a__geq#(X,Y)

and R consists of:

r1: a__minus(|0|(),Y) -> |0|()
r2: a__minus(s(X),s(Y)) -> a__minus(X,Y)
r3: a__geq(X,|0|()) -> true()
r4: a__geq(|0|(),s(Y)) -> false()
r5: a__geq(s(X),s(Y)) -> a__geq(X,Y)
r6: a__div(|0|(),s(Y)) -> |0|()
r7: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),|0|())
r8: a__if(true(),X,Y) -> mark(X)
r9: a__if(false(),X,Y) -> mark(Y)
r10: mark(minus(X1,X2)) -> a__minus(X1,X2)
r11: mark(geq(X1,X2)) -> a__geq(X1,X2)
r12: mark(div(X1,X2)) -> a__div(mark(X1),X2)
r13: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
r14: mark(|0|()) -> |0|()
r15: mark(s(X)) -> s(mark(X))
r16: mark(true()) -> true()
r17: mark(false()) -> false()
r18: a__minus(X1,X2) -> minus(X1,X2)
r19: a__geq(X1,X2) -> geq(X1,X2)
r20: a__div(X1,X2) -> div(X1,X2)
r21: a__if(X1,X2,X3) -> if(X1,X2,X3)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        a__geq#_A(x1,x2) = max{0, x1 - 2, x2 - 2}
        s_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      a__geq# = s
    
    partial status:
  
      pi(a__geq#) = []
      pi(s) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.