YES
We show the termination of the TRS R:

  |app'|(|app'|(minus(),x),|0|()) -> x
  |app'|(|app'|(minus(),|app'|(s(),x)),|app'|(s(),y)) -> |app'|(|app'|(minus(),x),y)
  |app'|(|app'|(quot(),|0|()),|app'|(s(),y)) -> |0|()
  |app'|(|app'|(quot(),|app'|(s(),x)),|app'|(s(),y)) -> |app'|(s(),|app'|(|app'|(quot(),|app'|(|app'|(minus(),x),y)),|app'|(s(),y)))
  |app'|(|app'|(le(),|0|()),y) -> true()
  |app'|(|app'|(le(),|app'|(s(),x)),|0|()) -> false()
  |app'|(|app'|(le(),|app'|(s(),x)),|app'|(s(),y)) -> |app'|(|app'|(le(),x),y)
  |app'|(|app'|(app(),nil()),y) -> y
  |app'|(|app'|(app(),|app'|(|app'|(add(),n),x)),y) -> |app'|(|app'|(add(),n),|app'|(|app'|(app(),x),y))
  |app'|(|app'|(low(),n),nil()) -> nil()
  |app'|(|app'|(low(),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(|app'|(if_low(),|app'|(|app'|(le(),m),n)),n),|app'|(|app'|(add(),m),x))
  |app'|(|app'|(|app'|(if_low(),true()),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(add(),m),|app'|(|app'|(low(),n),x))
  |app'|(|app'|(|app'|(if_low(),false()),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(low(),n),x)
  |app'|(|app'|(high(),n),nil()) -> nil()
  |app'|(|app'|(high(),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(|app'|(if_high(),|app'|(|app'|(le(),m),n)),n),|app'|(|app'|(add(),m),x))
  |app'|(|app'|(|app'|(if_high(),true()),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(high(),n),x)
  |app'|(|app'|(|app'|(if_high(),false()),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(add(),m),|app'|(|app'|(high(),n),x))
  |app'|(quicksort(),nil()) -> nil()
  |app'|(quicksort(),|app'|(|app'|(add(),n),x)) -> |app'|(|app'|(app(),|app'|(quicksort(),|app'|(|app'|(low(),n),x))),|app'|(|app'|(add(),n),|app'|(quicksort(),|app'|(|app'|(high(),n),x))))
  |app'|(|app'|(map(),f),nil()) -> nil()
  |app'|(|app'|(map(),f),|app'|(|app'|(add(),x),xs)) -> |app'|(|app'|(add(),|app'|(f,x)),|app'|(|app'|(map(),f),xs))
  |app'|(|app'|(filter(),f),nil()) -> nil()
  |app'|(|app'|(filter(),f),|app'|(|app'|(add(),x),xs)) -> |app'|(|app'|(|app'|(|app'|(filter2(),|app'|(f,x)),f),x),xs)
  |app'|(|app'|(|app'|(|app'|(filter2(),true()),f),x),xs) -> |app'|(|app'|(add(),x),|app'|(|app'|(filter(),f),xs))
  |app'|(|app'|(|app'|(|app'|(filter2(),false()),f),x),xs) -> |app'|(|app'|(filter(),f),xs)

Instead we prove termination of the uncurried system:

  |app'|(add_0(),y) -> add_1(y)
  |app'|(add_1(x1),y) -> add_2(x1,y)
  |app'|(app_0(),y) -> app_1(y)
  |app'|(app_1(x1),y) -> app_2(x1,y)
  |app'|(filter_0(),y) -> filter_1(y)
  |app'|(filter_1(x1),y) -> filter_2(x1,y)
  |app'|(filter2_0(),y) -> filter2_1(y)
  |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
  |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
  |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
  |app'|(high_0(),y) -> high_1(y)
  |app'|(high_1(x1),y) -> high_2(x1,y)
  |app'|(if_high_0(),y) -> if_high_1(y)
  |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
  |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
  |app'|(if_low_0(),y) -> if_low_1(y)
  |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
  |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
  |app'|(le_0(),y) -> le_1(y)
  |app'|(le_1(x1),y) -> le_2(x1,y)
  |app'|(low_0(),y) -> low_1(y)
  |app'|(low_1(x1),y) -> low_2(x1,y)
  |app'|(map_0(),y) -> map_1(y)
  |app'|(map_1(x1),y) -> map_2(x1,y)
  |app'|(minus_0(),y) -> minus_1(y)
  |app'|(minus_1(x1),y) -> minus_2(x1,y)
  |app'|(quicksort_0(),y) -> quicksort_1(y)
  |app'|(quot_0(),y) -> quot_1(y)
  |app'|(quot_1(x1),y) -> quot_2(x1,y)
  |app'|(s_0(),y) -> s_1(y)
  minus_2(x,|0|_0()) -> x
  minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
  quot_2(|0|_0(),s_1(y)) -> |0|_0()
  quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
  le_2(|0|_0(),y) -> true_0()
  le_2(s_1(x),|0|_0()) -> false_0()
  le_2(s_1(x),s_1(y)) -> le_2(x,y)
  app_2(nil_0(),y) -> y
  app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
  low_2(n,nil_0()) -> nil_0()
  low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
  if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
  if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
  high_2(n,nil_0()) -> nil_0()
  high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
  if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
  if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
  quicksort_1(nil_0()) -> nil_0()
  quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
  filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
  filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)


-- SCC decomposition.

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

p1: |app'|#(app_1(x1),y) -> app_2#(x1,y)
p2: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p3: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: |app'|#(high_1(x1),y) -> high_2#(x1,y)
p5: |app'|#(if_high_2(x1,x2),y) -> if_high_3#(x1,x2,y)
p6: |app'|#(if_low_2(x1,x2),y) -> if_low_3#(x1,x2,y)
p7: |app'|#(le_1(x1),y) -> le_2#(x1,y)
p8: |app'|#(low_1(x1),y) -> low_2#(x1,y)
p9: |app'|#(map_1(x1),y) -> map_2#(x1,y)
p10: |app'|#(minus_1(x1),y) -> minus_2#(x1,y)
p11: |app'|#(quicksort_0(),y) -> quicksort_1#(y)
p12: |app'|#(quot_1(x1),y) -> quot_2#(x1,y)
p13: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p14: quot_2#(s_1(x),s_1(y)) -> quot_2#(minus_2(x,y),s_1(y))
p15: quot_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p16: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)
p17: app_2#(add_2(n,x),y) -> app_2#(x,y)
p18: low_2#(n,add_2(m,x)) -> if_low_3#(le_2(m,n),n,add_2(m,x))
p19: low_2#(n,add_2(m,x)) -> le_2#(m,n)
p20: if_low_3#(true_0(),n,add_2(m,x)) -> low_2#(n,x)
p21: if_low_3#(false_0(),n,add_2(m,x)) -> low_2#(n,x)
p22: high_2#(n,add_2(m,x)) -> if_high_3#(le_2(m,n),n,add_2(m,x))
p23: high_2#(n,add_2(m,x)) -> le_2#(m,n)
p24: if_high_3#(true_0(),n,add_2(m,x)) -> high_2#(n,x)
p25: if_high_3#(false_0(),n,add_2(m,x)) -> high_2#(n,x)
p26: quicksort_1#(add_2(n,x)) -> app_2#(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
p27: quicksort_1#(add_2(n,x)) -> quicksort_1#(low_2(n,x))
p28: quicksort_1#(add_2(n,x)) -> low_2#(n,x)
p29: quicksort_1#(add_2(n,x)) -> quicksort_1#(high_2(n,x))
p30: quicksort_1#(add_2(n,x)) -> high_2#(n,x)
p31: map_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p32: map_2#(f,add_2(x,xs)) -> map_2#(f,xs)
p33: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p34: filter_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p35: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p36: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p9, p31, p32, p33, p34, p35, p36}
  {p27, p29}
  {p17}
  {p22, p24, p25}
  {p18, p20, p21}
  {p16}
  {p14}
  {p13}


-- Reduction pair.

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

p1: map_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p2: |app'|#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,add_2(x,xs)) -> map_2#(f,xs)
p4: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: filter_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p7: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p8: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p9: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r48, r50, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{2, x1 - 89, x2 - 61}
            add_2_A(x1,x2) = max{60, x1 + 4, x2}
            |app'|#_A(x1,x2) = max{2, x1 - 93, x2 - 59}
            map_1_A(x1) = max{92, x1 + 4}
            filter2_3_A(x1,x2,x3) = max{x1 - 4, x2 + 9, x3 + 40}
            filter2_4#_A(x1,x2,x3,x4) = max{2, x2 - 84, x3 - 55, x4 - 59}
            false_0_A = 63
            filter_2#_A(x1,x2) = max{2, x1 - 84, x2 - 59}
            filter_1_A(x1) = max{92, x1 + 9}
            |app'|_A(x1,x2) = max{65, x1 - 34}
            true_0_A = 10
            quicksort_1_A(x1) = max{69, x1 + 7}
            nil_0_A = 5
            map_2_A(x1,x2) = max{59, x1 - 30, x2 + 7}
            filter_2_A(x1,x2) = max{67, x1 - 32, x2 + 7}
            map_2#!_A(x1,x2) = max{86, x1 - 2}
            add_2!_A(x1,x2) = 85
            |app'|#!_A(x1,x2) = max{2, x1 - 5}
            map_1!_A(x1) = max{152, x1}
            filter2_3!_A(x1,x2,x3) = max{x2 + 150, x3 - 1}
            filter2_4#!_A(x1,x2,x3,x4) = x2 + 3
            false_0!_A = 150
            filter_2#!_A(x1,x2) = x1 + 3
            filter_1!_A(x1) = 150
            |app'|!_A(x1,x2) = max{x1 + 152, x2 + 146}
            true_0!_A = 86
            quicksort_1!_A(x1) = max{0, x1 - 67}
            nil_0!_A = 0
            map_2!_A(x1,x2) = max{1, x1 - 1, x2 - 6}
            filter_2!_A(x1,x2) = 149
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{1, x1 - 131, x2 - 35}
            add_2_A(x1,x2) = max{34, x1 + 18, x2 + 13}
            |app'|#_A(x1,x2) = max{1, x1 - 132, x2 - 35}
            map_1_A(x1) = max{33, x1 + 1}
            filter2_3_A(x1,x2,x3) = max{x2 + 92, x3 + 92}
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 40, x3 - 40, x4 - 35}
            false_0_A = 30
            filter_2#_A(x1,x2) = max{1, x1 - 40, x2 - 35}
            filter_1_A(x1) = max{131, x1 + 124}
            |app'|_A(x1,x2) = 32
            true_0_A = 2
            quicksort_1_A(x1) = x1
            nil_0_A = 83
            map_2_A(x1,x2) = max{x1 - 14, x2 + 15}
            filter_2_A(x1,x2) = max{227, x1 - 14, x2 + 2}
            map_2#!_A(x1,x2) = max{7, x2 - 12}
            add_2!_A(x1,x2) = max{95, x1 + 92, x2 - 132}
            |app'|#!_A(x1,x2) = max{8, x2 + 6}
            map_1!_A(x1) = max{49, x1 - 122}
            filter2_3!_A(x1,x2,x3) = max{x1 + 94, x2 - 148}
            filter2_4#!_A(x1,x2,x3,x4) = max{x3 + 16, x4 + 11}
            false_0!_A = 96
            filter_2#!_A(x1,x2) = max{10, x2 - 1}
            filter_1!_A(x1) = 0
            |app'|!_A(x1,x2) = max{54, x1 - 34, x2 + 51}
            true_0!_A = 56
            quicksort_1!_A(x1) = max{52, x1 + 14}
            nil_0!_A = 49
            map_2!_A(x1,x2) = max{50, x1 - 122, x2 - 131}
            filter_2!_A(x1,x2) = max{97, x1 - 34, x2 - 84}
        
  
    partial status:
  
      pi(map_2#) = []
      pi(add_2) = []
      pi(|app'|#) = []
      pi(map_1) = []
      pi(filter2_3) = [2]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = []
      pi(|app'|) = [1, 2]
      pi(true_0) = []
      pi(quicksort_1) = []
      pi(nil_0) = []
      pi(map_2) = [1, 2]
      pi(filter_2) = []

The following rules are removed:

  p1, p2, p3, p4, p5, p6, p7, p8, p9

Then no dependency pair remains.

-- Reduction pair.

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

p1: quicksort_1#(add_2(n,x)) -> quicksort_1#(high_2(n,x))
p2: quicksort_1#(add_2(n,x)) -> quicksort_1#(low_2(n,x))

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r35, r36, r37, r40, r41, r42, r43, r44, r45, r46, r47, r50

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quicksort_1#_A(x1) = 0
            add_2_A(x1,x2) = 14
            high_2_A(x1,x2) = max{160, x1 - 16}
            low_2_A(x1,x2) = x1 + 14
            le_2_A(x1,x2) = x2 + 149
            |0|_0_A = 183
            true_0_A = 4
            s_1_A(x1) = max{863, x1 + 518}
            false_0_A = 332
            nil_0_A = 3
            if_low_3_A(x1,x2,x3) = x2 + 14
            if_high_3_A(x1,x2,x3) = max{160, x1 - 331, x2 - 16}
            map_2_A(x1,x2) = max{13, x1 - 14, x2 - 15}
            quicksort_1#!_A(x1) = 0
            add_2!_A(x1,x2) = 177
            high_2!_A(x1,x2) = 177
            low_2!_A(x1,x2) = max{182, x1 - 1}
            le_2!_A(x1,x2) = max{181, x2 - 2}
            |0|_0!_A = 181
            true_0!_A = 0
            s_1!_A(x1) = x1 + 173
            false_0!_A = 180
            nil_0!_A = 183
            if_low_3!_A(x1,x2,x3) = max{181, x2 - 1, x3 + 168}
            if_high_3!_A(x1,x2,x3) = max{177, x3 - 15}
            map_2!_A(x1,x2) = 184
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quicksort_1#_A(x1) = 0
            add_2_A(x1,x2) = max{42, x2 + 22}
            high_2_A(x1,x2) = max{16, x2 + 1}
            low_2_A(x1,x2) = x2 + 18
            le_2_A(x1,x2) = 5
            |0|_0_A = 11
            true_0_A = 1
            s_1_A(x1) = max{13, x1}
            false_0_A = 3
            nil_0_A = 44
            if_low_3_A(x1,x2,x3) = max{41, x3 + 18}
            if_high_3_A(x1,x2,x3) = max{24, x3 + 1}
            map_2_A(x1,x2) = max{44, x1 - 10}
            quicksort_1#!_A(x1) = max{0, x1 - 17}
            add_2!_A(x1,x2) = 8
            high_2!_A(x1,x2) = 8
            low_2!_A(x1,x2) = 48
            le_2!_A(x1,x2) = 7
            |0|_0!_A = 48
            true_0!_A = 29
            s_1!_A(x1) = 49
            false_0!_A = 1
            nil_0!_A = 29
            if_low_3!_A(x1,x2,x3) = 48
            if_high_3!_A(x1,x2,x3) = max{8, x1 + 2}
            map_2!_A(x1,x2) = max{29, x1 - 31, x2 - 45}
        
  
    partial status:
  
      pi(quicksort_1#) = []
      pi(add_2) = [2]
      pi(high_2) = [2]
      pi(low_2) = [1, 2]
      pi(le_2) = [2]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_low_3) = [2, 3]
      pi(if_high_3) = [3]
      pi(map_2) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

p1: app_2#(add_2(n,x),y) -> app_2#(x,y)

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app_2#_A(x1,x2) = max{0, x1 - 38}
            add_2_A(x1,x2) = max{x1 - 215, x2 + 37}
            app_2#!_A(x1,x2) = max{3, x1 - 88, x2 - 1}
            add_2!_A(x1,x2) = max{2, x1 - 3, x2 - 88}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app_2#_A(x1,x2) = max{0, x1 - 6}
            add_2_A(x1,x2) = max{x1 - 63, x2 + 5}
            app_2#!_A(x1,x2) = max{5, x2 - 1}
            add_2!_A(x1,x2) = max{4, x1 - 344, x2 - 79}
        
  
    partial status:
  
      pi(app_2#) = [1]
      pi(add_2) = [2]

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: high_2#(n,add_2(m,x)) -> if_high_3#(le_2(m,n),n,add_2(m,x))
p2: if_high_3#(false_0(),n,add_2(m,x)) -> high_2#(n,x)
p3: if_high_3#(true_0(),n,add_2(m,x)) -> high_2#(n,x)

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r35, r36, r37, r44

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            high_2#_A(x1,x2) = max{0, x1 - 4, x2 - 32}
            add_2_A(x1,x2) = max{30, x1 - 129, x2 + 28}
            if_high_3#_A(x1,x2,x3) = max{0, x1 - 11, x2 - 4, x3 - 33}
            le_2_A(x1,x2) = max{10, x1 - 150, x2 - 16}
            false_0_A = 10
            true_0_A = 10
            |0|_0_A = 15
            s_1_A(x1) = max{17, x1 + 15}
            high_2_A(x1,x2) = max{10, x1 - 33, x2 - 33}
            nil_0_A = 0
            high_2#!_A(x1,x2) = max{13, x2 - 13}
            add_2!_A(x1,x2) = max{25, x1 + 24, x2 - 11}
            if_high_3#!_A(x1,x2,x3) = max{14, x3 - 40}
            le_2!_A(x1,x2) = 18
            false_0!_A = 1
            true_0!_A = 4
            |0|_0!_A = 14
            s_1!_A(x1) = max{18, x1 + 15}
            high_2!_A(x1,x2) = max{5, x1 - 164}
            nil_0!_A = 4
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            high_2#_A(x1,x2) = max{0, x1 - 6, x2 - 8}
            add_2_A(x1,x2) = max{6, x1 - 60, x2 + 3}
            if_high_3#_A(x1,x2,x3) = max{0, x1 - 75, x2 - 6, x3 - 11}
            le_2_A(x1,x2) = max{74, x1 - 10, x2 - 10}
            false_0_A = 4
            true_0_A = 0
            |0|_0_A = 8
            s_1_A(x1) = max{9, x1 + 7}
            high_2_A(x1,x2) = max{69, x1 - 26, x2 - 26}
            nil_0_A = 25
            high_2#!_A(x1,x2) = max{1, x1 - 104}
            add_2!_A(x1,x2) = max{68, x1 - 169, x2 - 71}
            if_high_3#!_A(x1,x2,x3) = max{68, x3 - 109}
            le_2!_A(x1,x2) = max{72, x1 - 62}
            false_0!_A = 69
            true_0!_A = 3
            |0|_0!_A = 4
            s_1!_A(x1) = x1 + 70
            high_2!_A(x1,x2) = max{x1 - 198, x2 + 61}
            nil_0!_A = 3
        
  
    partial status:
  
      pi(high_2#) = []
      pi(add_2) = [2]
      pi(if_high_3#) = []
      pi(le_2) = [1]
      pi(false_0) = []
      pi(true_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(high_2) = []
      pi(nil_0) = []

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_3#(true_0(),n,add_2(m,x)) -> low_2#(n,x)
p2: low_2#(n,add_2(m,x)) -> if_low_3#(le_2(m,n),n,add_2(m,x))
p3: if_low_3#(false_0(),n,add_2(m,x)) -> low_2#(n,x)

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r35, r36, r37, r40, r44

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_low_3#_A(x1,x2,x3) = max{0, x1 - 10, x2 - 238, x3 - 38}
            true_0_A = 9
            add_2_A(x1,x2) = max{8, x1 - 317, x2 + 5}
            low_2#_A(x1,x2) = max{0, x1 - 238, x2 - 33}
            le_2_A(x1,x2) = max{9, x1 - 340, x2 - 228}
            false_0_A = 0
            |0|_0_A = 227
            s_1_A(x1) = max{339, x1 + 227}
            low_2_A(x1,x2) = max{35, x1 - 239, x2 - 34}
            nil_0_A = 6
            high_2_A(x1,x2) = max{28, x1 - 239, x2 - 34}
            if_low_3#!_A(x1,x2,x3) = max{6, x3 - 369}
            true_0!_A = 1
            add_2!_A(x1,x2) = max{7, x2 - 362}
            low_2#!_A(x1,x2) = max{6, x2 - 368}
            le_2!_A(x1,x2) = max{5, x1 - 328, x2 - 604}
            false_0!_A = 10
            |0|_0!_A = 2
            s_1!_A(x1) = 9
            low_2!_A(x1,x2) = max{4, x1 - 605, x2 - 361}
            nil_0!_A = 3
            high_2!_A(x1,x2) = max{7, x1 - 603, x2 - 7}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_low_3#_A(x1,x2,x3) = max{0, x1 - 12, x2 - 22, x3 - 38}
            true_0_A = 7
            add_2_A(x1,x2) = max{x1 - 45, x2 + 18}
            low_2#_A(x1,x2) = max{0, x1 - 22, x2 - 20}
            le_2_A(x1,x2) = max{11, x1 - 53, x2 - 53}
            false_0_A = 5
            |0|_0_A = 10
            s_1_A(x1) = x1 + 6
            low_2_A(x1,x2) = max{73, x1 - 17, x2 - 27}
            nil_0_A = 18
            high_2_A(x1,x2) = max{28, x1 - 27, x2 - 21}
            if_low_3#!_A(x1,x2,x3) = 3
            true_0!_A = 14
            add_2!_A(x1,x2) = x1 + 5
            low_2#!_A(x1,x2) = 3
            le_2!_A(x1,x2) = max{15, x1 - 66}
            false_0!_A = 2
            |0|_0!_A = 15
            s_1!_A(x1) = 1
            low_2!_A(x1,x2) = max{12, x1 - 66, x2 - 21}
            nil_0!_A = 8
            high_2!_A(x1,x2) = max{9, x1 - 68, x2 - 21}
        
  
    partial status:
  
      pi(if_low_3#) = [3]
      pi(true_0) = []
      pi(add_2) = [1, 2]
      pi(low_2#) = [2]
      pi(le_2) = []
      pi(false_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(low_2) = [1]
      pi(nil_0) = []
      pi(high_2) = [1]

The following rules are removed:

  p1, p3



-- SCC decomposition.

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

p1: low_2#(n,add_2(m,x)) -> if_low_3#(le_2(m,n),n,add_2(m,x))

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le_2#_A(x1,x2) = max{0, x1 - 5, x2 - 5}
            s_1_A(x1) = x1 + 4
            le_2#!_A(x1,x2) = max{6, x1 - 239, x2 - 5}
            s_1!_A(x1) = max{5, x1 - 239}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le_2#_A(x1,x2) = max{0, x1 - 5, x2 - 5}
            s_1_A(x1) = x1 + 4
            le_2#!_A(x1,x2) = max{0, x1 - 5}
            s_1!_A(x1) = max{9, x1 - 15}
        
  
    partial status:
  
      pi(le_2#) = [2]
      pi(s_1) = [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: quot_2#(s_1(x),s_1(y)) -> quot_2#(minus_2(x,y),s_1(y))

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r31, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quot_2#_A(x1,x2) = max{0, x1 - 6, x2 - 5}
            s_1_A(x1) = x1 + 7
            minus_2_A(x1,x2) = max{8, x1, x2 - 8}
            |0|_0_A = 3
            quot_2#!_A(x1,x2) = max{10, x2 + 8}
            s_1!_A(x1) = max{25, x1 + 16}
            minus_2!_A(x1,x2) = max{9, x1 - 462, x2 - 461}
            |0|_0!_A = 26
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quot_2#_A(x1,x2) = max{0, x1 - 39, x2 - 59}
            s_1_A(x1) = max{37, x1 + 21}
            minus_2_A(x1,x2) = x1 + 18
            |0|_0_A = 21
            quot_2#!_A(x1,x2) = max{1, x1 - 19}
            s_1!_A(x1) = max{23, x1 + 21}
            minus_2!_A(x1,x2) = max{38, x1 - 331, x2 - 37}
            |0|_0!_A = 24
        
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = [1]
      pi(minus_2) = [1, 2]
      pi(|0|_0) = []

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_2#(s_1(x),s_1(y)) -> minus_2#(x,y)

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(high_0(),y) -> high_1(y)
r12: |app'|(high_1(x1),y) -> high_2(x1,y)
r13: |app'|(if_high_0(),y) -> if_high_1(y)
r14: |app'|(if_high_1(x1),y) -> if_high_2(x1,y)
r15: |app'|(if_high_2(x1,x2),y) -> if_high_3(x1,x2,y)
r16: |app'|(if_low_0(),y) -> if_low_1(y)
r17: |app'|(if_low_1(x1),y) -> if_low_2(x1,y)
r18: |app'|(if_low_2(x1,x2),y) -> if_low_3(x1,x2,y)
r19: |app'|(le_0(),y) -> le_1(y)
r20: |app'|(le_1(x1),y) -> le_2(x1,y)
r21: |app'|(low_0(),y) -> low_1(y)
r22: |app'|(low_1(x1),y) -> low_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(minus_0(),y) -> minus_1(y)
r26: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r27: |app'|(quicksort_0(),y) -> quicksort_1(y)
r28: |app'|(quot_0(),y) -> quot_1(y)
r29: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: minus_2(x,|0|_0()) -> x
r32: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r33: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r34: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: low_2(n,nil_0()) -> nil_0()
r41: low_2(n,add_2(m,x)) -> if_low_3(le_2(m,n),n,add_2(m,x))
r42: if_low_3(true_0(),n,add_2(m,x)) -> add_2(m,low_2(n,x))
r43: if_low_3(false_0(),n,add_2(m,x)) -> low_2(n,x)
r44: high_2(n,nil_0()) -> nil_0()
r45: high_2(n,add_2(m,x)) -> if_high_3(le_2(m,n),n,add_2(m,x))
r46: if_high_3(true_0(),n,add_2(m,x)) -> high_2(n,x)
r47: if_high_3(false_0(),n,add_2(m,x)) -> add_2(m,high_2(n,x))
r48: quicksort_1(nil_0()) -> nil_0()
r49: quicksort_1(add_2(n,x)) -> app_2(quicksort_1(low_2(n,x)),add_2(n,quicksort_1(high_2(n,x))))
r50: map_2(f,nil_0()) -> nil_0()
r51: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r52: filter_2(f,nil_0()) -> nil_0()
r53: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r54: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r55: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus_2#_A(x1,x2) = max{0, x1 - 11, x2 - 11}
            s_1_A(x1) = x1 + 10
            minus_2#!_A(x1,x2) = max{1, x1 - 11, x2 - 11}
            s_1!_A(x1) = max{0, x1 - 1198}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus_2#_A(x1,x2) = max{0, x1 - 5, x2 - 5}
            s_1_A(x1) = x1 + 4
            minus_2#!_A(x1,x2) = max{15, x1 - 1339, x2 - 1340}
            s_1!_A(x1) = max{14, x1 - 1339}
        
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.