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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_qs#_A(x1,x2,x3,x4) = 3
          false_A() = 4
          quicksort#_A(x1) = 3
          isempty_A(x1) = x1 + 2
          low_A(x1,x2) = x2
          head_A(x1) = x1 + 4
          tail_A(x1) = x1 + 4
          high_A(x1,x2) = x2 + 5
          le_A(x1,x2) = x1 + x2 + 1
          |0|_A() = 5
          true_A() = 1
          s_A(x1) = x1 + 5
          nil_A() = 0
          add_A(x1,x2) = x1 + x2 + 6
          if_low_A(x1,x2,x3) = x3
          if_high_A(x1,x2,x3) = x3 + 5
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_qs#_A(x1,x2,x3,x4) = max{83, x1 - 48, x2 + 21, x3 - 1, x4 + 21}
          false_A = 137
          quicksort#_A(x1) = max{83, x1 + 21}
          isempty_A(x1) = max{69, x1 + 66}
          low_A(x1,x2) = x2 + 21
          head_A(x1) = max{0, x1 - 67}
          tail_A(x1) = max{38, x1 - 25}
          high_A(x1,x2) = max{62, x2 + 23}
          le_A(x1,x2) = max{62, x1 - 1}
          |0|_A = 0
          true_A = 1
          s_A(x1) = max{138, x1 + 69}
          nil_A = 21
          add_A(x1,x2) = max{x1 + 70, x2 + 74}
          if_low_A(x1,x2,x3) = max{94, x1 - 138, x3 + 21}
          if_high_A(x1,x2,x3) = max{94, x1 - 1, x3 + 23}
          if_qs#!_A(x1,x2,x3,x4) = max{x1 - 70, x2 + 3, x3 + 1, x4 + 2}
          false!_A = 65
          quicksort#!_A(x1) = max{66, x1 + 2}
          isempty!_A(x1) = max{1, x1 - 20}
          low!_A(x1,x2) = max{137, x1 - 2, x2 - 36}
          head!_A(x1) = max{67, x1 - 2}
          tail!_A(x1) = max{66, x1 - 2}
          high!_A(x1,x2) = max{137, x2 - 56}
          le!_A(x1,x2) = max{0, x1 - 5}
          |0|!_A = 66
          true!_A = 0
          s!_A(x1) = max{66, x1 - 6}
          nil!_A = 69
          add!_A(x1,x2) = max{0, x1 - 4}
          if_low!_A(x1,x2,x3) = max{0, x1 - 2, x3 - 37}
          if_high!_A(x1,x2,x3) = max{0, x3 - 56}
  
    partial status:
  
      pi(if_qs#) = []
      pi(false) = []
      pi(quicksort#) = []
      pi(isempty) = []
      pi(low) = [2]
      pi(head) = []
      pi(tail) = []
      pi(high) = [2]
      pi(le) = [1]
      pi(|0|) = []
      pi(true) = []
      pi(s) = [1]
      pi(nil) = []
      pi(add) = [1]
      pi(if_low) = [3]
      pi(if_high) = [3]

The following rules are removed:

  p2, p3



-- SCC decomposition.

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

p1: 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:

  (no SCCs)

-- 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, r10

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_high#_A(x1,x2,x3) = x2 + x3 + 6
          false_A() = 12
          add_A(x1,x2) = x2 + 6
          high#_A(x1,x2) = x1 + x2 + 11
          le_A(x1,x2) = x2 + 16
          true_A() = 10
          |0|_A() = 13
          s_A(x1) = x1 + 17
          high_A(x1,x2) = 15
          nil_A() = 2
  
    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 - 12, x2 - 5, x3 - 14}
          false_A = 5
          add_A(x1,x2) = max{x1 - 28, x2 + 12}
          high#_A(x1,x2) = max{0, x1 - 5, x2 - 14}
          le_A(x1,x2) = max{11, x1 - 41, x2 - 3}
          true_A = 11
          |0|_A = 2
          s_A(x1) = max{40, x1 + 1}
          high_A(x1,x2) = max{49, x1 - 26, x2 - 26}
          nil_A = 25
          if_high#!_A(x1,x2,x3) = max{6, x1 - 4, x2 - 1, x3 - 13}
          false!_A = 6
          add!_A(x1,x2) = 0
          high#!_A(x1,x2) = max{6, x2 - 16}
          le!_A(x1,x2) = max{6, x1 - 45}
          true!_A = 5
          |0|!_A = 7
          s!_A(x1) = max{6, x1 - 45}
          high!_A(x1,x2) = max{4, x1 - 76, x2 - 88}
          nil!_A = 3
  
    partial status:
  
      pi(if_high#) = [2]
      pi(false) = []
      pi(add) = []
      pi(high#) = [1, 2]
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(s) = [1]
      pi(high) = []
      pi(nil) = []

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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_low#_A(x1,x2,x3) = x2 + x3 + 1
          false_A() = 7
          add_A(x1,x2) = x2 + 8
          low#_A(x1,x2) = x1 + x2 + 6
          le_A(x1,x2) = x2 + 13
          true_A() = 12
          |0|_A() = 13
          s_A(x1) = x1 + 14
  
    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 - 67, x3 - 72}
          false_A = 1
          add_A(x1,x2) = max{x1 - 58, x2 + 14}
          low#_A(x1,x2) = max{0, x1 - 67, x2 - 72}
          le_A(x1,x2) = max{1, x1 - 128, x2 - 128}
          true_A = 1
          |0|_A = 127
          s_A(x1) = max{127, x1 + 126}
          if_low#!_A(x1,x2,x3) = max{8, x1 - 67, x2 - 1, x3 - 20}
          false!_A = 5
          add!_A(x1,x2) = 0
          low#!_A(x1,x2) = max{4, x2 - 7}
          le!_A(x1,x2) = max{6, x1 - 128}
          true!_A = 3
          |0|!_A = 6
          s!_A(x1) = max{6, x1 - 128}
  
    partial status:
  
      pi(if_low#) = [2]
      pi(false) = []
      pi(add) = []
      pi(low#) = [1, 2]
      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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          le#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          le#_A(x1,x2) = 0
          s_A(x1) = max{0, x1 - 1}
          le#!_A(x1,x2) = max{59, x1 - 1, x2 - 1}
          s!_A(x1) = max{58, x1 - 2}
  
    partial status:
  
      pi(le#) = [2]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = 1
          add_A(x1,x2) = x1 + 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = x1
          add_A(x1,x2) = max{39, x1 + 34, x2 + 3}
          app#!_A(x1,x2) = max{1, x1 - 1}
          add!_A(x1,x2) = max{0, x1 - 20, x2 - 23}
  
    partial status:
  
      pi(app#) = []
      pi(add) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.