YES
We show the termination of the TRS R:

  le(|0|(),y) -> true()
  le(s(x),|0|()) -> false()
  le(s(x),s(y)) -> le(x,y)
  app(nil(),y) -> y
  app(add(n,x),y) -> add(n,app(x,y))
  low(n,nil()) -> nil()
  low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
  if_low(true(),n,add(m,x)) -> add(m,low(n,x))
  if_low(false(),n,add(m,x)) -> low(n,x)
  high(n,nil()) -> nil()
  high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
  if_high(true(),n,add(m,x)) -> high(n,x)
  if_high(false(),n,add(m,x)) -> add(m,high(n,x))
  head(add(n,x)) -> n
  tail(add(n,x)) -> x
  isempty(nil()) -> true()
  isempty(add(n,x)) -> false()
  quicksort(x) -> if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
  if_qs(true(),x,n,y) -> nil()
  if_qs(false(),x,n,y) -> app(quicksort(x),add(n,quicksort(y)))


-- SCC decomposition.

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

p1: le#(s(x),s(y)) -> le#(x,y)
p2: app#(add(n,x),y) -> app#(x,y)
p3: low#(n,add(m,x)) -> if_low#(le(m,n),n,add(m,x))
p4: low#(n,add(m,x)) -> le#(m,n)
p5: if_low#(true(),n,add(m,x)) -> low#(n,x)
p6: if_low#(false(),n,add(m,x)) -> low#(n,x)
p7: high#(n,add(m,x)) -> if_high#(le(m,n),n,add(m,x))
p8: high#(n,add(m,x)) -> le#(m,n)
p9: if_high#(true(),n,add(m,x)) -> high#(n,x)
p10: if_high#(false(),n,add(m,x)) -> high#(n,x)
p11: quicksort#(x) -> if_qs#(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
p12: quicksort#(x) -> isempty#(x)
p13: quicksort#(x) -> low#(head(x),tail(x))
p14: quicksort#(x) -> head#(x)
p15: quicksort#(x) -> tail#(x)
p16: quicksort#(x) -> high#(head(x),tail(x))
p17: if_qs#(false(),x,n,y) -> app#(quicksort(x),add(n,quicksort(y)))
p18: if_qs#(false(),x,n,y) -> quicksort#(x)
p19: if_qs#(false(),x,n,y) -> quicksort#(y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: app(nil(),y) -> y
r5: app(add(n,x),y) -> add(n,app(x,y))
r6: low(n,nil()) -> nil()
r7: low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
r8: if_low(true(),n,add(m,x)) -> add(m,low(n,x))
r9: if_low(false(),n,add(m,x)) -> low(n,x)
r10: high(n,nil()) -> nil()
r11: high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
r12: if_high(true(),n,add(m,x)) -> high(n,x)
r13: if_high(false(),n,add(m,x)) -> add(m,high(n,x))
r14: head(add(n,x)) -> n
r15: tail(add(n,x)) -> x
r16: isempty(nil()) -> true()
r17: isempty(add(n,x)) -> false()
r18: quicksort(x) -> if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
r19: if_qs(true(),x,n,y) -> nil()
r20: if_qs(false(),x,n,y) -> app(quicksort(x),add(n,quicksort(y)))

The estimated dependency graph contains the following SCCs:

  {p11, p18, p19}
  {p7, p9, p10}
  {p3, p5, p6}
  {p1}
  {p2}


-- Reduction pair.

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

p1: if_qs#(false(),x,n,y) -> quicksort#(y)
p2: quicksort#(x) -> if_qs#(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
p3: if_qs#(false(),x,n,y) -> quicksort#(x)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: app(nil(),y) -> y
r5: app(add(n,x),y) -> add(n,app(x,y))
r6: low(n,nil()) -> nil()
r7: low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
r8: if_low(true(),n,add(m,x)) -> add(m,low(n,x))
r9: if_low(false(),n,add(m,x)) -> low(n,x)
r10: high(n,nil()) -> nil()
r11: high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
r12: if_high(true(),n,add(m,x)) -> high(n,x)
r13: if_high(false(),n,add(m,x)) -> add(m,high(n,x))
r14: head(add(n,x)) -> n
r15: tail(add(n,x)) -> x
r16: isempty(nil()) -> true()
r17: isempty(add(n,x)) -> false()
r18: quicksort(x) -> if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
r19: if_qs(true(),x,n,y) -> nil()
r20: if_qs(false(),x,n,y) -> app(quicksort(x),add(n,quicksort(y)))

The set of usable rules consists of

  r1, r2, r3, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_qs#_A(x1,x2,x3,x4) = max{x1 + 2, x2 + 7, x3 + 1, x4 + 7}
        false_A = 41
        quicksort#_A(x1) = max{42, x1 + 7}
        isempty_A(x1) = max{6, x1 + 4}
        low_A(x1,x2) = max{19, x2}
        head_A(x1) = max{8, x1 + 1}
        tail_A(x1) = max{0, x1 - 9}
        high_A(x1,x2) = max{21, x2 + 2}
        le_A(x1,x2) = 41
        |0|_A = 0
        true_A = 41
        s_A(x1) = 0
        nil_A = 39
        add_A(x1,x2) = max{39, x1, x2 + 18}
        if_low_A(x1,x2,x3) = max{39, x3}
        if_high_A(x1,x2,x3) = max{40, x1, x3 + 2}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_qs#_A(x1,x2,x3,x4) = 1
          false_A = 64
          quicksort#_A(x1) = 1
          isempty_A(x1) = max{56, x1 + 24}
          low_A(x1,x2) = max{54, x2}
          head_A(x1) = x1 + 24
          tail_A(x1) = max{55, x1}
          high_A(x1,x2) = max{33, x2}
          le_A(x1,x2) = max{0, x1 - 22}
          |0|_A = 150
          true_A = 57
          s_A(x1) = x1 + 149
          nil_A = 33
          add_A(x1,x2) = max{40, x1 - 24, x2}
          if_low_A(x1,x2,x3) = max{54, x3}
          if_high_A(x1,x2,x3) = max{40, x1 - 31, x3}
          if_qs#!_A(x1,x2,x3,x4) = max{x2 + 28, x3 + 84, x4 + 28}
          false!_A = 68
          quicksort#!_A(x1) = max{83, x1 + 27}
          isempty!_A(x1) = x1 + 27
          low!_A(x1,x2) = 2
          head!_A(x1) = max{28, x1 - 1}
          tail!_A(x1) = max{83, x1 - 1}
          high!_A(x1,x2) = max{5, x2 - 55}
          le!_A(x1,x2) = max{3, x1 - 80}
          |0|!_A = 0
          true!_A = 4
          s!_A(x1) = max{4, x1 - 81}
          nil!_A = 1
          add!_A(x1,x2) = 0
          if_low!_A(x1,x2,x3) = 2
          if_high!_A(x1,x2,x3) = max{1, x1 - 58, x3 - 95}
  
    partial status:
  
      pi(if_qs#) = []
      pi(false) = []
      pi(quicksort#) = [1]
      pi(isempty) = []
      pi(low) = []
      pi(head) = []
      pi(tail) = []
      pi(high) = []
      pi(le) = []
      pi(|0|) = []
      pi(true) = []
      pi(s) = []
      pi(nil) = []
      pi(add) = []
      pi(if_low) = []
      pi(if_high) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: if_high#(false(),n,add(m,x)) -> high#(n,x)
p2: high#(n,add(m,x)) -> if_high#(le(m,n),n,add(m,x))
p3: if_high#(true(),n,add(m,x)) -> high#(n,x)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: app(nil(),y) -> y
r5: app(add(n,x),y) -> add(n,app(x,y))
r6: low(n,nil()) -> nil()
r7: low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
r8: if_low(true(),n,add(m,x)) -> add(m,low(n,x))
r9: if_low(false(),n,add(m,x)) -> low(n,x)
r10: high(n,nil()) -> nil()
r11: high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
r12: if_high(true(),n,add(m,x)) -> high(n,x)
r13: if_high(false(),n,add(m,x)) -> add(m,high(n,x))
r14: head(add(n,x)) -> n
r15: tail(add(n,x)) -> x
r16: isempty(nil()) -> true()
r17: isempty(add(n,x)) -> false()
r18: quicksort(x) -> if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
r19: if_qs(true(),x,n,y) -> nil()
r20: if_qs(false(),x,n,y) -> app(quicksort(x),add(n,quicksort(y)))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_high#_A(x1,x2,x3) = max{20, x2 - 42}
        false_A = 0
        add_A(x1,x2) = max{0, x1 - 15, x2 - 17}
        high#_A(x1,x2) = max{20, x1 - 42}
        le_A(x1,x2) = max{19, x1 - 41, x2 - 41}
        true_A = 9
        |0|_A = 40
        s_A(x1) = max{20, x1 + 1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_high#_A(x1,x2,x3) = max{0, x1 - 2, x2 - 5, x3 - 12}
          false_A = 1
          add_A(x1,x2) = max{x1 + 6, x2 + 10}
          high#_A(x1,x2) = max{0, x1 - 5, x2 - 12}
          le_A(x1,x2) = max{1, x1 - 21, x2 - 21}
          true_A = 1
          |0|_A = 20
          s_A(x1) = x1 + 12
          if_high#!_A(x1,x2,x3) = max{8, x3 - 4}
          false!_A = 12
          add!_A(x1,x2) = max{14, x1 - 146, x2 - 64}
          high#!_A(x1,x2) = max{1, x2 - 1}
          le!_A(x1,x2) = max{10, x1 - 1, x2 - 21}
          true!_A = 18
          |0|!_A = 19
          s!_A(x1) = max{13, x1 - 22}
  
    partial status:
  
      pi(if_high#) = []
      pi(false) = []
      pi(add) = []
      pi(high#) = []
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(s) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: if_low#(false(),n,add(m,x)) -> low#(n,x)
p2: low#(n,add(m,x)) -> if_low#(le(m,n),n,add(m,x))
p3: if_low#(true(),n,add(m,x)) -> low#(n,x)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: app(nil(),y) -> y
r5: app(add(n,x),y) -> add(n,app(x,y))
r6: low(n,nil()) -> nil()
r7: low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
r8: if_low(true(),n,add(m,x)) -> add(m,low(n,x))
r9: if_low(false(),n,add(m,x)) -> low(n,x)
r10: high(n,nil()) -> nil()
r11: high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
r12: if_high(true(),n,add(m,x)) -> high(n,x)
r13: if_high(false(),n,add(m,x)) -> add(m,high(n,x))
r14: head(add(n,x)) -> n
r15: tail(add(n,x)) -> x
r16: isempty(nil()) -> true()
r17: isempty(add(n,x)) -> false()
r18: quicksort(x) -> if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
r19: if_qs(true(),x,n,y) -> nil()
r20: if_qs(false(),x,n,y) -> app(quicksort(x),add(n,quicksort(y)))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_low#_A(x1,x2,x3) = max{7, x2 - 41}
        false_A = 2
        add_A(x1,x2) = max{6, x1 - 12, x2 - 12}
        low#_A(x1,x2) = max{7, x1 - 41}
        le_A(x1,x2) = max{8, x1 - 40, x2 - 40}
        true_A = 5
        |0|_A = 39
        s_A(x1) = max{7, x1 + 1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_low#_A(x1,x2,x3) = max{0, x1 - 2, x2 - 3, x3 - 30}
          false_A = 1
          add_A(x1,x2) = max{28, x1 + 6, x2 + 13}
          low#_A(x1,x2) = max{0, x1 - 3, x2 - 30}
          le_A(x1,x2) = max{1, x1 - 23, x2 - 23}
          true_A = 1
          |0|_A = 22
          s_A(x1) = x1
          if_low#!_A(x1,x2,x3) = max{11, x3 - 2}
          false!_A = 28
          add!_A(x1,x2) = max{30, x1 - 77, x2 - 90}
          low#!_A(x1,x2) = max{3, x2 - 1}
          le!_A(x1,x2) = max{28, x1 - 222, x2 - 211}
          true!_A = 2
          |0|!_A = 29
          s!_A(x1) = max{28, x1 - 223}
  
    partial status:
  
      pi(if_low#) = []
      pi(false) = []
      pi(add) = []
      pi(low#) = []
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(s) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: le#(s(x),s(y)) -> le#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: app(nil(),y) -> y
r5: app(add(n,x),y) -> add(n,app(x,y))
r6: low(n,nil()) -> nil()
r7: low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
r8: if_low(true(),n,add(m,x)) -> add(m,low(n,x))
r9: if_low(false(),n,add(m,x)) -> low(n,x)
r10: high(n,nil()) -> nil()
r11: high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
r12: if_high(true(),n,add(m,x)) -> high(n,x)
r13: if_high(false(),n,add(m,x)) -> add(m,high(n,x))
r14: head(add(n,x)) -> n
r15: tail(add(n,x)) -> x
r16: isempty(nil()) -> true()
r17: isempty(add(n,x)) -> false()
r18: quicksort(x) -> if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
r19: if_qs(true(),x,n,y) -> nil()
r20: if_qs(false(),x,n,y) -> app(quicksort(x),add(n,quicksort(y)))

The set of usable rules consists of

  r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le#_A(x1,x2) = 0
        s_A(x1) = max{2, x1 - 4}
        isempty_A(x1) = max{6, x1 - 11}
        add_A(x1,x2) = max{7, x2 - 12}
        false_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          le#_A(x1,x2) = max{x1, x2}
          s_A(x1) = x1 + 5
          isempty_A(x1) = max{37, x1 - 16}
          add_A(x1,x2) = max{15, x1 - 81, x2 - 15}
          false_A = 4
          le#!_A(x1,x2) = max{3, x1 - 1, x2 - 6}
          s!_A(x1) = max{2, x1 - 81}
          isempty!_A(x1) = max{2, x1 - 59}
          add!_A(x1,x2) = max{6, x1 - 255, x2 - 127}
          false!_A = 1
  
    partial status:
  
      pi(le#) = []
      pi(s) = []
      pi(isempty) = []
      pi(add) = []
      pi(false) = []

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: app#(add(n,x),y) -> app#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: app(nil(),y) -> y
r5: app(add(n,x),y) -> add(n,app(x,y))
r6: low(n,nil()) -> nil()
r7: low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
r8: if_low(true(),n,add(m,x)) -> add(m,low(n,x))
r9: if_low(false(),n,add(m,x)) -> low(n,x)
r10: high(n,nil()) -> nil()
r11: high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
r12: if_high(true(),n,add(m,x)) -> high(n,x)
r13: if_high(false(),n,add(m,x)) -> add(m,high(n,x))
r14: head(add(n,x)) -> n
r15: tail(add(n,x)) -> x
r16: isempty(nil()) -> true()
r17: isempty(add(n,x)) -> false()
r18: quicksort(x) -> if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
r19: if_qs(true(),x,n,y) -> nil()
r20: if_qs(false(),x,n,y) -> app(quicksort(x),add(n,quicksort(y)))

The set of usable rules consists of

  r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{4, x1 - 4}
        add_A(x1,x2) = max{9, x1 - 27, x2 + 3}
        isempty_A(x1) = max{13, x1 - 37}
        false_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = 0
          add_A(x1,x2) = max{3, x1 - 19, x2 - 20}
          isempty_A(x1) = max{37, x1 - 21}
          false_A = 1
          app#!_A(x1,x2) = 0
          add!_A(x1,x2) = max{4, x1 - 3, x2 - 11}
          isempty!_A(x1) = max{2, x1 - 21}
          false!_A = 1
  
    partial status:
  
      pi(app#) = []
      pi(add) = [2]
      pi(isempty) = []
      pi(false) = []

The following rules are removed:

  p1

Then no dependency pair remains.