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

  r1, r5, r7, r11, r12, r13, r15, r16, r19, r21, r25, r28, r30, r31, r32, r33, r35, r36, r37, r40, r44, r45, r46, r47, r48, r50, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{6, x2 - 1}
        add_2_A(x1,x2) = max{x1 + 7, x2}
        |app'|#_A(x1,x2) = x2 + 6
        map_1_A(x1) = x1
        filter2_3_A(x1,x2,x3) = max{x1 - 21, x2 - 21, x3 + 20}
        filter2_4#_A(x1,x2,x3,x4) = max{5, x4 + 1}
        false_0_A = 20
        filter_2#_A(x1,x2) = max{4, x2 + 1}
        filter_1_A(x1) = max{3, x1 - 21}
        |app'|_A(x1,x2) = x2 + 20
        true_0_A = 20
        add_0_A = 1
        add_1_A(x1) = max{12, x1 - 21}
        filter_0_A = 1
        filter2_0_A = 1
        filter2_1_A(x1) = 0
        high_0_A = 1
        high_1_A(x1) = max{10, x1 - 21}
        high_2_A(x1,x2) = x2
        if_high_0_A = 1
        if_high_1_A(x1) = max{5, x1 - 21}
        if_high_2_A(x1,x2) = max{6, x1 - 21}
        if_high_3_A(x1,x2,x3) = max{7, x3}
        if_low_0_A = 1
        if_low_1_A(x1) = max{12, x1 - 21}
        le_0_A = 1
        le_1_A(x1) = max{10, x1 - 21}
        low_0_A = 11
        low_1_A(x1) = 10
        minus_0_A = 1
        minus_1_A(x1) = max{9, x1 + 1}
        quot_0_A = 1
        quot_1_A(x1) = max{12, x1 + 3}
        s_0_A = 1
        s_1_A(x1) = x1
        minus_2_A(x1,x2) = max{x1 + 11, x2 - 1}
        |0|_0_A = 1
        quot_2_A(x1,x2) = max{2, x1 - 12}
        le_2_A(x1,x2) = max{20, x1 - 2, x2 - 10}
        low_2_A(x1,x2) = max{x1 + 15, x2 - 10}
        nil_0_A = 9
        quicksort_1_A(x1) = max{14, x1 + 7}
        map_2_A(x1,x2) = 10
        filter_2_A(x1,x2) = max{10, x1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{29, x1 - 154, x2 - 19}
          add_2_A(x1,x2) = max{37, x1 + 10, x2 + 18}
          |app'|#_A(x1,x2) = max{29, x1 - 154, x2 - 9}
          map_1_A(x1) = max{153, x1 + 1}
          filter2_3_A(x1,x2,x3) = max{x1 - 89, x2 + 1, x3 + 29}
          filter2_4#_A(x1,x2,x3,x4) = max{29, x2 - 153, x3 - 125, x4 - 9}
          false_0_A = 29
          filter_2#_A(x1,x2) = max{29, x1 - 153, x2 - 9}
          filter_1_A(x1) = max{2, x1 + 1}
          |app'|_A(x1,x2) = max{140, x1 - 3, x2 + 29}
          true_0_A = 88
          add_0_A = 2
          add_1_A(x1) = x1 + 4
          filter_0_A = 2
          filter2_0_A = 2
          filter2_1_A(x1) = max{2, x1 - 30}
          high_0_A = 2
          high_1_A(x1) = max{2, x1 - 1}
          high_2_A(x1,x2) = max{19, x2}
          if_high_0_A = 2
          if_high_1_A(x1) = max{2, x1 - 31}
          if_high_2_A(x1,x2) = max{2, x1 - 34, x2 - 1}
          if_high_3_A(x1,x2,x3) = max{36, x3}
          if_low_0_A = 2
          if_low_1_A(x1) = max{2, x1 - 83}
          le_0_A = 2
          le_1_A(x1) = max{30, x1 - 1}
          low_0_A = 2
          low_1_A(x1) = max{30, x1 + 1}
          minus_0_A = 2
          minus_1_A(x1) = max{30, x1 + 29}
          quot_0_A = 2
          quot_1_A(x1) = x1 + 3
          s_0_A = 2
          s_1_A(x1) = x1 + 29
          minus_2_A(x1,x2) = max{30, x1}
          |0|_0_A = 92
          quot_2_A(x1,x2) = x1
          le_2_A(x1,x2) = max{87, x1 - 4}
          low_2_A(x1,x2) = max{38, x1 - 116, x2}
          nil_0_A = 17
          quicksort_1_A(x1) = max{141, x1 + 28}
          map_2_A(x1,x2) = max{150, x1 - 19, x2 + 28}
          filter_2_A(x1,x2) = max{47, x1 - 20, x2 + 28}
          map_2#!_A(x1,x2) = 0
          add_2!_A(x1,x2) = 0
          |app'|#!_A(x1,x2) = 0
          map_1!_A(x1) = 0
          filter2_3!_A(x1,x2,x3) = max{56, x1 - 29, x2 - 1, x3 - 31}
          filter2_4#!_A(x1,x2,x3,x4) = max{18, x4 - 1}
          false_0!_A = 58
          filter_2#!_A(x1,x2) = max{0, x2 - 18}
          filter_1!_A(x1) = 1
          |app'|!_A(x1,x2) = 53
          true_0!_A = 0
          add_0!_A = 1
          add_1!_A(x1) = max{0, x1 - 30}
          filter_0!_A = 2
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{54, x1 - 31}
          high_0!_A = 1
          high_1!_A(x1) = max{0, x1 - 1}
          high_2!_A(x1,x2) = max{57, x2 - 44}
          if_high_0!_A = 53
          if_high_1!_A(x1) = max{54, x1 - 32}
          if_high_2!_A(x1,x2) = max{0, x1 - 31, x2 - 1}
          if_high_3!_A(x1,x2,x3) = max{57, x1 - 30, x3 - 45}
          if_low_0!_A = 1
          if_low_1!_A(x1) = max{2, x1 - 91}
          le_0!_A = 1
          le_1!_A(x1) = max{54, x1 - 95}
          low_0!_A = 1
          low_1!_A(x1) = max{54, x1 - 1}
          minus_0!_A = 1
          minus_1!_A(x1) = max{54, x1 - 95}
          quot_0!_A = 55
          quot_1!_A(x1) = 54
          s_0!_A = 55
          s_1!_A(x1) = max{54, x1 - 95}
          minus_2!_A(x1,x2) = max{55, x1 - 94, x2 - 30}
          |0|_0!_A = 54
          quot_2!_A(x1,x2) = 53
          le_2!_A(x1,x2) = max{59, x1 - 94, x2 - 93}
          low_2!_A(x1,x2) = max{55, x1 - 94, x2 - 104}
          nil_0!_A = 24
          quicksort_1!_A(x1) = max{56, x1 + 8}
          map_2!_A(x1,x2) = 54
          filter_2!_A(x1,x2) = max{52, x1 + 10, x2 - 38}
  
    partial status:
  
      pi(map_2#) = []
      pi(add_2) = [2]
      pi(|app'|#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [3]
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = []
      pi(|app'|) = [2]
      pi(true_0) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(high_2) = [2]
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_high_3) = [3]
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(minus_2) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(le_2) = []
      pi(low_2) = [1]
      pi(nil_0) = []
      pi(quicksort_1) = []
      pi(map_2) = []
      pi(filter_2) = [1]

The following rules are removed:

  p4, p5, p6, p7, p8, p9



-- SCC decomposition.

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)

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:

  {p1, p2, p3}


-- 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)

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:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = 8
        add_2_A(x1,x2) = max{x1 - 40, x2 + 15}
        |app'|#_A(x1,x2) = 8
        map_1_A(x1) = max{36, x1 - 38}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x1 - 1, x2 - 192}
          add_2_A(x1,x2) = max{191, x1 + 19, x2 + 1}
          |app'|#_A(x1,x2) = max{0, x1 - 18, x2 - 192}
          map_1_A(x1) = x1 + 17
          map_2#!_A(x1,x2) = max{13, x1 - 1, x2 - 419}
          add_2!_A(x1,x2) = max{x1 - 418, x2 + 11}
          |app'|#!_A(x1,x2) = max{12, x1 - 3, x2 - 418}
          map_1!_A(x1) = max{5, x1 - 450}
  
    partial status:
  
      pi(map_2#) = []
      pi(add_2) = [2]
      pi(|app'|#) = []
      pi(map_1) = []

The following rules are removed:

  p1, p2



-- SCC decomposition.

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

p1: map_2#(f,add_2(x,xs)) -> map_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:

  {p1}


-- Reduction pair.

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

p1: map_2#(f,add_2(x,xs)) -> map_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

  r52

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = 1
        add_2_A(x1,x2) = max{9, x1 - 35, x2 - 36}
        filter_2_A(x1,x2) = max{11, x1 - 40, x2 - 36}
        nil_0_A = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x2 - 5}
          add_2_A(x1,x2) = max{15, x1 - 110, x2 + 4}
          filter_2_A(x1,x2) = max{266, x1 - 159, x2 - 287}
          nil_0_A = 33
          map_2#!_A(x1,x2) = max{9, x2 - 5}
          add_2!_A(x1,x2) = max{8, x1 - 398, x2 - 267}
          filter_2!_A(x1,x2) = max{10, x1 - 358, x2 - 356}
          nil_0!_A = 0
  
    partial status:
  
      pi(map_2#) = []
      pi(add_2) = []
      pi(filter_2) = []
      pi(nil_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: 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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quicksort_1#_A(x1) = max{10, x1 - 23}
        add_2_A(x1,x2) = max{x1 + 34, x2 + 15}
        high_2_A(x1,x2) = max{14, x2 + 2}
        low_2_A(x1,x2) = max{21, x2 + 5}
        le_2_A(x1,x2) = max{35, x1 - 40}
        |0|_0_A = 35
        true_0_A = 1
        s_1_A(x1) = max{36, x1 + 8}
        false_0_A = 13
        nil_0_A = 19
        if_low_3_A(x1,x2,x3) = max{35, x1 - 36, x3 + 5}
        if_high_3_A(x1,x2,x3) = max{27, x1 - 36, x3 + 2}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          quicksort_1#_A(x1) = max{0, x1 - 26}
          add_2_A(x1,x2) = max{6, x1 - 24, x2}
          high_2_A(x1,x2) = max{25, x1 - 24, x2}
          low_2_A(x1,x2) = max{25, x1 - 24, x2}
          le_2_A(x1,x2) = max{2, x1 - 50, x2 - 62}
          |0|_0_A = 61
          true_0_A = 11
          s_1_A(x1) = x1 + 69
          false_0_A = 2
          nil_0_A = 1
          if_low_3_A(x1,x2,x3) = max{25, x1 - 25, x2 - 24, x3}
          if_high_3_A(x1,x2,x3) = max{25, x1 - 12, x2 - 24, x3}
          quicksort_1#!_A(x1) = max{5, x1 - 26}
          add_2!_A(x1,x2) = 8
          high_2!_A(x1,x2) = max{209, x1 - 1, x2 - 17}
          low_2!_A(x1,x2) = max{6, x1 + 2, x2 - 28}
          le_2!_A(x1,x2) = max{5, x1 - 70, x2 - 70}
          |0|_0!_A = 2
          true_0!_A = 1
          s_1!_A(x1) = max{4, x1 - 133}
          false_0!_A = 1
          nil_0!_A = 10
          if_low_3!_A(x1,x2,x3) = max{5, x1 - 3, x2 + 1, x3 - 29}
          if_high_3!_A(x1,x2,x3) = max{9, x1 - 12, x2 - 2, x3 - 24}
  
    partial status:
  
      pi(quicksort_1#) = []
      pi(add_2) = []
      pi(high_2) = []
      pi(low_2) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_low_3) = []
      pi(if_high_3) = []

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:
  
      max/plus interpretation on natural numbers:
      
        app_2#_A(x1,x2) = x2 + 8
        add_2_A(x1,x2) = max{6, x1 - 35, x2 - 45}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app_2#_A(x1,x2) = max{0, x1 - 2}
          add_2_A(x1,x2) = max{10, x1 - 85, x2 + 1}
          app_2#!_A(x1,x2) = max{1, x1 - 8}
          add_2!_A(x1,x2) = max{4, x1 - 117, x2 - 31}
  
    partial status:
  
      pi(app_2#) = []
      pi(add_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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        high_2#_A(x1,x2) = max{9, x1 - 34}
        add_2_A(x1,x2) = max{9, x1 - 51, x2 - 51}
        if_high_3#_A(x1,x2,x3) = max{9, x2 - 34}
        le_2_A(x1,x2) = max{16, x1 - 33, x2 - 33}
        false_0_A = 1
        true_0_A = 10
        |0|_0_A = 31
        s_1_A(x1) = max{32, x1 + 15}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          high_2#_A(x1,x2) = max{0, x1 - 8, x2 - 12}
          add_2_A(x1,x2) = max{10, x1 - 74, x2 + 2}
          if_high_3#_A(x1,x2,x3) = max{0, x1 - 3, x2 - 8, x3 - 12}
          le_2_A(x1,x2) = max{2, x1 - 83, x2 - 5}
          false_0_A = 2
          true_0_A = 2
          |0|_0_A = 4
          s_1_A(x1) = max{4, x1}
          high_2#!_A(x1,x2) = max{80, x2 + 76}
          add_2!_A(x1,x2) = max{86, x1 - 128, x2 - 180}
          if_high_3#!_A(x1,x2,x3) = max{79, x3 + 75}
          le_2!_A(x1,x2) = max{92, x1 - 5, x2 - 5}
          false_0!_A = 81
          true_0!_A = 87
          |0|_0!_A = 88
          s_1!_A(x1) = max{90, x1 - 116}
  
    partial status:
  
      pi(high_2#) = []
      pi(add_2) = []
      pi(if_high_3#) = []
      pi(le_2) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(|0|_0) = []
      pi(s_1) = [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_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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_low_3#_A(x1,x2,x3) = max{2, x2 - 77, x3 - 116}
        true_0_A = 16
        add_2_A(x1,x2) = max{82, x2 + 3}
        low_2#_A(x1,x2) = max{2, x1 - 77, x2 - 116}
        le_2_A(x1,x2) = max{x1 + 233, x2 - 77}
        false_0_A = 5
        |0|_0_A = 17
        s_1_A(x1) = max{76, x1 + 18}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_low_3#_A(x1,x2,x3) = max{0, x1 - 17, x2 - 3, x3 - 40}
          true_0_A = 3
          add_2_A(x1,x2) = max{38, x1 - 93, x2 + 18}
          low_2#_A(x1,x2) = max{0, x1 - 3, x2 - 40}
          le_2_A(x1,x2) = max{16, x1 - 116}
          false_0_A = 1
          |0|_0_A = 109
          s_1_A(x1) = max{109, x1 + 2}
          if_low_3#!_A(x1,x2,x3) = max{130, x1 - 2, x3 + 94}
          true_0!_A = 9
          add_2!_A(x1,x2) = max{150, x2 + 7}
          low_2#!_A(x1,x2) = max{131, x2 + 111}
          le_2!_A(x1,x2) = max{156, x1 - 110, x2 - 127}
          false_0!_A = 9
          |0|_0!_A = 156
          s_1!_A(x1) = max{155, x1 + 8}
  
    partial status:
  
      pi(if_low_3#) = []
      pi(true_0) = []
      pi(add_2) = [2]
      pi(low_2#) = []
      pi(le_2) = [1]
      pi(false_0) = []
      pi(|0|_0) = []
      pi(s_1) = [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_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:
  
      max/plus interpretation on natural numbers:
      
        le_2#_A(x1,x2) = 1
        s_1_A(x1) = max{7, x1 - 38}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          le_2#_A(x1,x2) = max{0, x1 - 11, x2 - 11}
          s_1_A(x1) = x1 + 10
          le_2#!_A(x1,x2) = max{5, x1 - 11, x2 - 4}
          s_1!_A(x1) = max{4, x1 - 152}
  
    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: 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:
  
      max/plus interpretation on natural numbers:
      
        quot_2#_A(x1,x2) = max{1, x1 - 9}
        s_1_A(x1) = max{16, x1 + 11}
        minus_2_A(x1,x2) = max{8, x1 + 2}
        |0|_0_A = 15
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          quot_2#_A(x1,x2) = max{0, x1 - 7, x2 - 7}
          s_1_A(x1) = max{5, x1 + 1}
          minus_2_A(x1,x2) = max{6, x1 + 1, x2 - 89}
          |0|_0_A = 3
          quot_2#!_A(x1,x2) = max{0, x2 - 91}
          s_1!_A(x1) = max{10, x1 - 131}
          minus_2!_A(x1,x2) = max{11, x1 - 130, x2 - 127}
          |0|_0!_A = 9
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = [1]
      pi(minus_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:
  
      max/plus interpretation on natural numbers:
      
        minus_2#_A(x1,x2) = 0
        s_1_A(x1) = max{4, x1 - 118}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus_2#_A(x1,x2) = max{0, x1 - 2, x2 - 2}
          s_1_A(x1) = max{4, x1 + 1}
          minus_2#!_A(x1,x2) = max{1, x1 - 2, x2 - 352}
          s_1!_A(x1) = max{4, x1 - 351}
  
    partial status:
  
      pi(minus_2#) = []
      pi(s_1) = []

The following rules are removed:

  p1

Then no dependency pair remains.