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, r2, r3, r7, r8, r11, r13, r14, r16, r17, r19, r21, r25, r28, r30, r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{7, x1 + 6}
        add_2_A(x1,x2) = max{8, x1 - 32, x2 - 37}
        |app'|#_A(x1,x2) = max{4, x1}
        map_1_A(x1) = x1 + 7
        filter2_3_A(x1,x2,x3) = max{x1 + 8, x2 + 6, x3 + 4}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 5
        false_0_A = 1
        filter_2#_A(x1,x2) = x1 + 5
        filter_1_A(x1) = max{8, x1 + 6}
        |app'|_A(x1,x2) = max{21, x1 - 9, x2 - 9}
        true_0_A = 3
        add_0_A = 8
        add_1_A(x1) = max{7, x1 - 10}
        app_0_A = 7
        app_1_A(x1) = max{8, x1 - 156}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 10}
        filter2_2_A(x1,x2) = max{3, x1 - 19, x2 - 9}
        high_0_A = 5
        high_1_A(x1) = max{6, x1 - 10}
        if_high_0_A = 6
        if_high_1_A(x1) = max{7, x1 - 10}
        if_high_2_A(x1,x2) = max{8, x1 - 19, x2 - 9}
        if_low_0_A = 6
        if_low_1_A(x1) = max{7, x1 - 9}
        if_low_2_A(x1,x2) = max{8, x1 - 116, x2 - 147}
        le_0_A = 8
        le_1_A(x1) = max{5, x1 - 10}
        low_0_A = 8
        low_1_A(x1) = 7
        minus_0_A = 7
        minus_1_A(x1) = max{8, x1 - 329}
        quot_0_A = 7
        quot_1_A(x1) = max{8, x1 - 10}
        s_0_A = 4
        s_1_A(x1) = max{5, x1 - 159}
        quot_2_A(x1,x2) = max{20, x1 - 179, x2 - 344}
        |0|_0_A = 4
        le_2_A(x1,x2) = max{6, x1 - 158, x2 - 119}
        low_2_A(x1,x2) = max{x1 - 118, x2 + 42}
        nil_0_A = 6
        high_2_A(x1,x2) = max{7, x1 + 5, x2 - 125}
        quicksort_1_A(x1) = max{132, x1 + 85}
        filter_2_A(x1,x2) = max{19, x1 - 32, x2 - 20}
  
    base order 2:
  
      precedence:
      
        filter_2 > |app'| > add_2 = map_1 = filter2_3 = filter_1 = if_high_2 = if_low_0 > add_1 = filter2_1 = filter2_2 = if_high_1 = if_low_2 = quicksort_1 > add_0 = filter2_0 = if_high_0 = if_low_1 = low_0 = low_2 = high_2 > app_1 = high_1 = low_1 = s_1 = quot_2 = le_2 = nil_0 > map_2# = app_0 = high_0 = le_1 = minus_1 = quot_1 = s_0 = |0|_0 > |app'|# = false_0 = true_0 = le_0 = minus_0 = quot_0 > filter2_4# = filter_2#
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(add_2) = []
      pi(|app'|#) = [1]
      pi(map_1) = []
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = [1]
      pi(|app'|) = []
      pi(true_0) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_2) = []

The following rules are removed:

  p2



-- 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: map_2#(f,add_2(x,xs)) -> map_2#(f,xs)
p3: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p6: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p7: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p8: 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 estimated dependency graph contains the following SCCs:

  {p2}
  {p3, p4, p5, p6, p7, p8}


-- 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

  r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r30, r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{x1 + 14, x2 + 30}
        add_2_A(x1,x2) = max{x1 - 16, x2 + 12}
        |app'|_A(x1,x2) = max{15, x1 - 34, x2 - 31}
        app_0_A = 6
        app_1_A(x1) = max{7, x1 - 32}
        filter_0_A = 13
        filter_1_A(x1) = max{14, x1 - 32}
        filter2_0_A = 1
        filter2_1_A(x1) = max{14, x1 - 32}
        filter2_2_A(x1,x2) = max{15, x1 - 67, x2 - 31}
        filter2_3_A(x1,x2,x3) = max{14, x1 - 101, x2 - 65, x3 - 31}
        high_0_A = 8
        high_1_A(x1) = max{9, x1 - 32}
        if_high_0_A = 1
        if_high_1_A(x1) = max{2, x1 - 32}
        if_high_2_A(x1,x2) = max{14, x1 - 67, x2 - 57}
        if_low_0_A = 33
        if_low_1_A(x1) = max{5, x1 - 32}
        if_low_2_A(x1,x2) = max{6, x1 - 66, x2 - 31}
        le_0_A = 3
        le_1_A(x1) = max{4, x1 - 32}
        low_0_A = 4
        low_1_A(x1) = max{5, x1 - 32}
        map_0_A = 33
        map_1_A(x1) = 14
        minus_0_A = 1
        minus_1_A(x1) = max{2, x1 - 224}
        quot_0_A = 1
        quot_1_A(x1) = max{2, x1 - 96}
        s_0_A = 3
        s_1_A(x1) = max{4, x1 - 127}
        quot_2_A(x1,x2) = max{5, x1 - 131, x2 - 263}
        |0|_0_A = 3
        le_2_A(x1,x2) = max{5, x1 - 126, x2 - 103}
        true_0_A = 2
        false_0_A = 2
        low_2_A(x1,x2) = max{6, x1 - 102, x2 - 109}
        nil_0_A = 0
        high_2_A(x1,x2) = max{10, x1 - 102, x2 - 109}
        quicksort_1_A(x1) = max{11, x1 - 23}
        filter_2_A(x1,x2) = max{16, x1 - 33, x2 - 14}
  
    base order 2:
  
      precedence:
      
        filter2_0 > filter2_1 = filter_2 > map_2# = |app'| = filter_0 = map_0 > add_2 = filter_1 = filter2_3 = high_0 = map_1 > filter2_2 = if_high_2 = if_low_2 = quicksort_1 > if_high_1 = if_low_1 = low_2 = high_2 > app_1 = high_1 = if_high_0 = if_low_0 = low_1 = le_2 = nil_0 > app_0 = le_1 = low_0 = s_1 = quot_2 > le_0 = quot_1 = s_0 = |0|_0 > minus_1 = quot_0 = true_0 = false_0 > minus_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(add_2) = [2]
      pi(|app'|) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_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: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p6: |app'|#(filter_1(x1),y) -> filter_2#(x1,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

  r1, r2, r3, r7, r8, r11, r13, r16, r17, r19, r21, r23, r25, r28, r30, r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        |app'|#_A(x1,x2) = max{2, x1 - 1}
        filter2_3_A(x1,x2,x3) = max{x1 + 8, x2 + 5, x3 + 5}
        filter2_4#_A(x1,x2,x3,x4) = max{3, x2 + 1}
        true_0_A = 1
        filter_2#_A(x1,x2) = max{3, x1 + 1}
        add_2_A(x1,x2) = max{9, x1 - 27, x2 - 27}
        |app'|_A(x1,x2) = max{26, x1 - 12, x2 - 12}
        false_0_A = 4
        filter_1_A(x1) = x1 + 5
        add_0_A = 7
        add_1_A(x1) = max{8, x1 - 13}
        app_0_A = 10
        app_1_A(x1) = max{11, x1 - 13}
        filter2_0_A = 2
        filter2_1_A(x1) = max{3, x1 - 13}
        filter2_2_A(x1,x2) = max{4, x1 - 25, x2 - 13}
        high_0_A = 10
        high_1_A(x1) = max{11, x1 - 13}
        if_high_0_A = 5
        if_high_1_A(x1) = max{6, x1 - 13}
        if_low_0_A = 5
        if_low_1_A(x1) = max{6, x1 - 12}
        if_low_2_A(x1,x2) = max{7, x1 - 104, x2 - 98}
        le_0_A = 5
        le_1_A(x1) = max{6, x1 - 96}
        low_0_A = 6
        low_1_A(x1) = max{7, x1 - 97}
        map_0_A = 10
        map_1_A(x1) = max{11, x1 - 27}
        minus_0_A = 11
        minus_1_A(x1) = max{13, x1 - 194}
        quot_0_A = 10
        quot_1_A(x1) = max{11, x1 - 13}
        s_0_A = 5
        s_1_A(x1) = max{6, x1 - 112}
        quot_2_A(x1,x2) = max{25, x1 - 94, x2 - 213}
        |0|_0_A = 5
        le_2_A(x1,x2) = max{7, x1 - 109, x2 - 111}
        low_2_A(x1,x2) = max{8, x1 - 110, x2 - 81}
        nil_0_A = 7
        high_2_A(x1,x2) = max{25, x1 - 27, x2 - 27}
        quicksort_1_A(x1) = max{13, x1 - 53}
        filter_2_A(x1,x2) = max{25, x1 - 18, x2 - 14}
  
    base order 2:
  
      precedence:
      
        |app'| > filter_2 > filter_1 = if_high_1 > filter2_4# = filter_2# > |app'|# = map_1 > filter2_3 = add_2 = filter2_1 = filter2_2 = if_high_0 = map_0 = quot_0 = high_2 = quicksort_1 > add_1 = filter2_0 = high_1 = if_low_2 = low_2 > add_0 = app_1 = high_0 = if_low_1 = low_1 = |0|_0 = le_2 = nil_0 > true_0 = false_0 = app_0 = if_low_0 = le_1 = low_0 > le_0 = s_1 = quot_2 > quot_1 = s_0 > minus_1 > minus_0
      
  
    partial status:
  
      pi(|app'|#) = []
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(true_0) = []
      pi(filter_2#) = [1]
      pi(add_2) = []
      pi(|app'|) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p3: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p5: |app'|#(filter_1(x1),y) -> filter_2#(x1,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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,add_2(x,xs)) -> |app'|#(f,x)
p3: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p4: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p5: 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 set of usable rules consists of

  r1, r2, r3, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r30, r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = max{8, x2 + 4}
        true_0_A = 6
        filter_2#_A(x1,x2) = max{8, x1 + 4}
        add_2_A(x1,x2) = max{19, x1 - 75, x2 - 26}
        |app'|#_A(x1,x2) = max{3, x1 - 13}
        filter_1_A(x1) = x1 + 21
        |app'|_A(x1,x2) = max{24, x1 + 5, x2 - 23}
        false_0_A = 13
        add_0_A = 1
        add_1_A(x1) = max{18, x1 - 23}
        app_0_A = 1
        app_1_A(x1) = max{7, x1 - 24}
        filter2_0_A = 21
        filter2_1_A(x1) = max{25, x1 - 58}
        filter2_2_A(x1,x2) = max{7, x1 - 57, x2 - 24}
        filter2_3_A(x1,x2,x3) = max{13, x1 - 52, x2 - 19, x3 - 23}
        high_0_A = 21
        high_1_A(x1) = max{25, x1 - 32}
        if_high_0_A = 1
        if_high_1_A(x1) = max{7, x1 - 94}
        if_high_2_A(x1,x2) = max{13, x1 - 93, x2 - 24}
        if_low_0_A = 1
        if_low_1_A(x1) = max{7, x1 - 24}
        if_low_2_A(x1,x2) = max{21, x1 - 25, x2 - 24}
        le_0_A = 1
        le_1_A(x1) = max{7, x1 - 196}
        low_0_A = 7
        low_1_A(x1) = max{13, x1 - 76}
        map_0_A = 1
        map_1_A(x1) = max{19, x1 - 69}
        minus_0_A = 1
        minus_1_A(x1) = max{7, x1 - 224}
        quot_0_A = 1
        quot_1_A(x1) = max{7, x1 - 24}
        s_0_A = 1
        s_1_A(x1) = max{15, x1 - 198}
        quot_2_A(x1,x2) = max{16, x1 - 20, x2 - 234}
        |0|_0_A = 14
        le_2_A(x1,x2) = max{16, x1 - 197, x2 - 76}
        low_2_A(x1,x2) = max{19, x1 - 75, x2 - 54}
        nil_0_A = 16
        high_2_A(x1,x2) = max{17, x1 - 27, x2 - 121}
        quicksort_1_A(x1) = max{18, x1 - 22}
        filter_2_A(x1,x2) = max{20, x1 + 6, x2 - 47}
  
    base order 2:
  
      precedence:
      
        filter_2 > filter_1 = filter2_0 > |app'|# = |app'| = filter2_1 = filter2_2 = filter2_3 = if_high_0 = low_0 > filter2_4# = filter_2# > add_2 = if_high_2 = map_1 = high_2 > add_1 = high_1 = if_high_1 = if_low_2 = map_0 = low_2 > add_0 = high_0 = if_low_1 = low_1 = le_2 = quicksort_1 > if_low_0 = le_1 = s_1 = quot_2 = nil_0 > app_1 = le_0 = quot_1 = s_0 = |0|_0 > true_0 = false_0 = app_0 = minus_1 = quot_0 > minus_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(add_2) = []
      pi(|app'|#) = []
      pi(filter_1) = [1]
      pi(|app'|) = [1]
      pi(false_0) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_2) = [1]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p2: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p3: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p4: 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:

  {p1, p3, p4}


-- Reduction pair.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p3: 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 set of usable rules consists of

  r1, r2, r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r30, r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = max{62, x3 + 10, x4 + 54}
        true_0_A = 70
        filter_2#_A(x1,x2) = x2 + 53
        add_2_A(x1,x2) = max{x1 + 10, x2 + 62}
        |app'|_A(x1,x2) = max{x1 - 21, x2 + 64}
        false_0_A = 65
        add_0_A = 1
        add_1_A(x1) = x1 + 43
        app_0_A = 1
        app_1_A(x1) = max{64, x1 - 1}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 86}
        filter2_0_A = 1
        filter2_1_A(x1) = max{19, x1 - 1}
        filter2_2_A(x1,x2) = max{20, x1 - 23, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{64, x1 - 44, x2 - 23, x3 - 1}
        high_0_A = 20
        high_1_A(x1) = max{19, x1 - 96}
        if_high_0_A = 1
        if_high_1_A(x1) = max{7, x1 - 1}
        if_high_2_A(x1,x2) = max{8, x1 - 50, x2 - 97}
        if_low_0_A = 19
        if_low_1_A(x1) = max{20, x1 - 1}
        if_low_2_A(x1,x2) = max{22, x1 - 72, x2 - 1}
        le_0_A = 1
        le_1_A(x1) = max{2, x1 - 52}
        low_0_A = 20
        low_1_A(x1) = max{17, x1 - 1}
        map_0_A = 1
        map_1_A(x1) = max{15, x1 - 42}
        minus_0_A = 22
        minus_1_A(x1) = max{63, x1 - 72}
        quot_0_A = 19
        quot_1_A(x1) = 20
        s_0_A = 1
        s_1_A(x1) = max{2, x1 - 145}
        quot_2_A(x1,x2) = max{72, x1 - 74, x2 + 64}
        |0|_0_A = 71
        le_2_A(x1,x2) = max{71, x1 - 72, x2 - 119}
        low_2_A(x1,x2) = max{9, x2}
        nil_0_A = 0
        high_2_A(x1,x2) = max{45, x1 - 118, x2}
        quicksort_1_A(x1) = max{65, x1 - 127}
        filter_2_A(x1,x2) = max{3, x1 - 108, x2 - 61}
  
    base order 2:
  
      precedence:
      
        quicksort_1 > nil_0 = filter_2 > |app'| = app_0 > app_1 = filter_1 = filter2_3 = if_high_0 > add_2 = filter_0 = filter2_2 = map_1 > filter2_4# = add_1 = filter2_1 = map_0 = high_2 > add_0 = filter2_0 = high_1 = if_low_2 = low_2 > high_0 = if_high_2 = if_low_1 = low_1 = le_2 > if_high_1 = if_low_0 = le_1 = low_0 = s_1 = quot_2 > le_0 = quot_1 = s_0 = |0|_0 > false_0 = minus_1 = quot_0 > filter_2# = minus_0 > true_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = [2]
      pi(add_2) = [1, 2]
      pi(|app'|) = [2]
      pi(false_0) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(low_2) = [2]
      pi(nil_0) = []
      pi(high_2) = [2]
      pi(quicksort_1) = []
      pi(filter_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p2: 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:

  (no SCCs)

-- 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

  r1, r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r25, r33, r35, r36, r37, r40, r41, r42, r43, r44, r45, r46, r47, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quicksort_1#_A(x1) = x1 + 59
        add_2_A(x1,x2) = max{x1 - 57, x2 + 69}
        high_2_A(x1,x2) = x2 + 10
        low_2_A(x1,x2) = x2 + 9
        |app'|_A(x1,x2) = max{53, x1 - 138, x2 - 186}
        add_0_A = 191
        add_1_A(x1) = max{52, x1 - 186}
        app_0_A = 137
        app_1_A(x1) = max{52, x1 - 187}
        filter_0_A = 51
        filter_1_A(x1) = max{52, x1 - 187}
        filter2_0_A = 1
        filter2_1_A(x1) = max{50, x1 - 187}
        filter2_2_A(x1,x2) = max{51, x1 - 326, x2 - 186}
        filter2_3_A(x1,x2,x3) = max{52, x1 - 465, x2 - 325, x3 - 187}
        high_0_A = 4
        high_1_A(x1) = max{5, x1 - 187}
        if_high_0_A = 52
        if_high_1_A(x1) = max{53, x1 - 187}
        if_high_2_A(x1,x2) = max{53, x1 - 325, x2 - 186}
        if_low_0_A = 137
        if_low_1_A(x1) = max{0, x1 - 187}
        if_low_2_A(x1,x2) = max{0, x1 - 326, x2 - 187}
        le_0_A = 1
        le_1_A(x1) = max{52, x1 - 187}
        le_2_A(x1,x2) = max{7, x1 - 326}
        low_0_A = 137
        low_1_A(x1) = max{0, x1 - 187}
        minus_0_A = 1
        minus_1_A(x1) = max{2, x1 - 187}
        quot_2_A(x1,x2) = max{52, x1 - 140, x2 - 186}
        |0|_0_A = 51
        s_1_A(x1) = max{95, x1 + 50}
        true_0_A = 1
        false_0_A = 5
        nil_0_A = 71
        if_low_3_A(x1,x2,x3) = max{78, x1 - 8, x3 + 9}
        if_high_3_A(x1,x2,x3) = max{74, x1 - 8, x3 + 10}
        filter_2_A(x1,x2) = max{72, x1 - 71, x2 - 128}
  
    base order 2:
  
      precedence:
      
        high_2 = low_2 > if_low_3 = if_high_3 = filter_2 > add_2 = |app'| = le_1 > quicksort_1# = add_1 = filter_1 = filter2_1 = filter2_2 = filter2_3 = high_1 = if_high_2 = if_low_2 = le_0 = le_2 = low_1 = quot_2 > add_0 = app_1 = filter_0 = filter2_0 = high_0 = if_high_1 = if_low_1 = low_0 > app_0 = if_high_0 = if_low_0 = minus_1 = s_1 = nil_0 > minus_0 = |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(quicksort_1#) = [1]
      pi(add_2) = []
      pi(high_2) = []
      pi(low_2) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(true_0) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_low_3) = [3]
      pi(if_high_3) = [3]
      pi(filter_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: quicksort_1#(add_2(n,x)) -> quicksort_1#(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 estimated dependency graph contains the following SCCs:

  {p1}


-- 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))

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, r3, r8, r13, r14, r19, r20, r21, r23, r25, r33, r35, r36, r37, r40, r44, r45, r46, r47, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quicksort_1#_A(x1) = x1 + 9
        add_2_A(x1,x2) = max{x1 - 7, x2 + 18}
        high_2_A(x1,x2) = x2 + 6
        |app'|_A(x1,x2) = max{5, x1 - 20, x2 - 61}
        add_0_A = 1
        add_1_A(x1) = max{4, x1 - 62}
        app_0_A = 1
        app_1_A(x1) = max{5, x1 - 62}
        filter2_1_A(x1) = max{21, x1 - 62}
        filter2_2_A(x1,x2) = max{5, x1 - 83, x2 - 62}
        if_high_0_A = 1
        if_high_1_A(x1) = max{2, x1 - 62}
        if_high_2_A(x1,x2) = max{0, x1 - 83}
        le_0_A = 1
        le_1_A(x1) = max{0, x1 - 62}
        le_2_A(x1,x2) = max{0, x1 - 128}
        low_0_A = 3
        low_1_A(x1) = max{4, x1 - 85}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 62}
        minus_0_A = 1
        minus_1_A(x1) = max{4, x1 - 62}
        quot_2_A(x1,x2) = max{2, x1 - 124, x2 - 211}
        |0|_0_A = 1
        s_1_A(x1) = max{127, x1 + 41}
        true_0_A = 0
        false_0_A = 0
        low_2_A(x1,x2) = max{0, x1 - 86, x2 - 60}
        nil_0_A = 0
        if_high_3_A(x1,x2,x3) = max{23, x1 - 41, x3 + 6}
        filter_2_A(x1,x2) = max{6, x1 - 19, x2 - 53}
  
    base order 2:
  
      precedence:
      
        quot_2 > add_0 = if_high_0 = filter_2 > |app'| > add_2 = high_2 = if_high_1 = le_1 = map_1 = if_high_3 > quicksort_1# = add_1 = app_1 = filter2_2 = if_high_2 = le_0 = map_0 = minus_1 > app_0 = filter2_1 = minus_0 > low_2 > le_2 = low_1 = nil_0 > low_0 = s_1 > |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(quicksort_1#) = [1]
      pi(add_2) = [2]
      pi(high_2) = [2]
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(if_high_3) = [3]
      pi(filter_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: 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

  r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app_2#_A(x1,x2) = max{x1 + 36, x2 + 12}
        add_2_A(x1,x2) = max{x1 - 24, x2 + 10}
        quot_2_A(x1,x2) = max{15, x1 - 43, x2 - 87}
        |0|_0_A = 3
        s_1_A(x1) = max{4, x1 - 39}
        le_2_A(x1,x2) = max{5, x1 - 38, x2 - 36}
        true_0_A = 1
        false_0_A = 2
        low_2_A(x1,x2) = max{16, x1 - 11, x2 - 13}
        nil_0_A = 0
        high_2_A(x1,x2) = max{6, x1 - 35, x2 - 12}
        quicksort_1_A(x1) = max{9, x1 - 41}
        filter_2_A(x1,x2) = max{13, x1 - 14, x2 - 24}
  
    base order 2:
  
      precedence:
      
        filter_2 > app_2# = add_2 = low_2 > high_2 > le_2 = quicksort_1 > quot_2 = s_1 = nil_0 > |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(app_2#) = [1]
      pi(add_2) = [2]
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_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

  r33, r35, r36, r37, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        high_2#_A(x1,x2) = max{5, x2 + 2}
        add_2_A(x1,x2) = max{x1 + 13, x2 + 9}
        if_high_3#_A(x1,x2,x3) = max{10, x1 + 3, x3 - 3}
        le_2_A(x1,x2) = x1 + 9
        false_0_A = 41
        true_0_A = 4
        quot_2_A(x1,x2) = max{82, x1 + 13, x2 - 83}
        |0|_0_A = 4
        s_1_A(x1) = x1 + 41
        filter_2_A(x1,x2) = max{81, x1 - 83, x2 - 68}
        nil_0_A = 13
  
    base order 2:
  
      precedence:
      
        high_2# = add_2 > if_high_3# = filter_2 = nil_0 > quot_2 = s_1 > le_2 = |0|_0 > true_0 > false_0
      
  
    partial status:
  
      pi(high_2#) = [2]
      pi(add_2) = [2]
      pi(if_high_3#) = [1]
      pi(le_2) = [1]
      pi(false_0) = []
      pi(true_0) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: if_high_3#(false_0(),n,add_2(m,x)) -> high_2#(n,x)
p2: 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 estimated dependency graph contains the following SCCs:

  (no SCCs)

-- 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

  r1, r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r30, r33, r35, r36, r37, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_low_3#_A(x1,x2,x3) = max{65, x1 + 4, x2 + 31, x3 + 33}
        true_0_A = 26
        add_2_A(x1,x2) = max{32, x1, x2 + 21}
        low_2#_A(x1,x2) = max{x1 + 31, x2 + 33}
        le_2_A(x1,x2) = max{x1 - 5, x2 + 27}
        false_0_A = 28
        |app'|_A(x1,x2) = max{x1 - 20, x2 + 37}
        add_0_A = 18
        add_1_A(x1) = max{19, x1 - 1}
        app_0_A = 18
        app_1_A(x1) = max{19, x1 - 1}
        filter_0_A = 1
        filter_1_A(x1) = max{19, x1 - 85}
        filter2_0_A = 19
        filter2_1_A(x1) = max{37, x1 - 1}
        filter2_2_A(x1,x2) = max{18, x1 - 22, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{19, x1 - 42, x2 - 22, x3 - 1}
        high_0_A = 22
        high_1_A(x1) = max{1, x1 - 1}
        if_high_0_A = 18
        if_high_1_A(x1) = max{19, x1 - 1}
        if_high_2_A(x1,x2) = max{18, x1 - 25, x2 - 3}
        if_low_0_A = 17
        if_low_1_A(x1) = max{18, x1 - 1}
        if_low_2_A(x1,x2) = max{19, x1 - 22, x2 - 1}
        le_0_A = 1
        le_1_A(x1) = max{19, x1 - 6}
        low_0_A = 2
        low_1_A(x1) = max{3, x1 - 1}
        map_0_A = 1
        map_1_A(x1) = max{37, x1 + 1}
        minus_0_A = 19
        minus_1_A(x1) = max{21, x1 - 16}
        quot_0_A = 18
        quot_1_A(x1) = max{19, x1 - 1}
        s_0_A = 35
        s_1_A(x1) = x1 + 35
        quot_2_A(x1,x2) = max{36, x1 - 74, x2 - 75}
        |0|_0_A = 25
        low_2_A(x1,x2) = max{4, x1 - 23, x2 + 2}
        nil_0_A = 1
        high_2_A(x1,x2) = max{24, x1 - 22, x2 - 51}
        quicksort_1_A(x1) = max{23, x1 - 73}
        filter_2_A(x1,x2) = max{31, x1 - 84, x2 - 96}
  
    base order 2:
  
      precedence:
      
        if_low_3# = low_2# > false_0 > le_2 > add_2 > high_2 = quicksort_1 > nil_0 = filter_2 > |app'| = filter_0 = high_0 = minus_1 > add_1 = app_0 = filter_1 = filter2_2 = filter2_3 = high_1 = if_high_2 = if_low_2 = le_1 = map_0 = minus_0 = quot_1 = s_0 = quot_2 = low_2 > add_0 = app_1 = filter2_1 = if_high_1 = if_low_1 = le_0 = low_1 = map_1 = quot_0 = s_1 > filter2_0 = if_high_0 = if_low_0 = low_0 = |0|_0 > true_0
      
  
    partial status:
  
      pi(if_low_3#) = [3]
      pi(true_0) = []
      pi(add_2) = [2]
      pi(low_2#) = [2]
      pi(le_2) = []
      pi(false_0) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(low_2) = [2]
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- 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))
p2: 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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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))
p2: 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

  r1, r3, r5, r7, r8, r11, r13, r14, r16, r17, r19, r28, r33, r35, r36, r37, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        low_2#_A(x1,x2) = max{7, x2 - 3}
        add_2_A(x1,x2) = max{51, x1 + 36, x2 + 27}
        if_low_3#_A(x1,x2,x3) = max{23, x3 - 28}
        le_2_A(x1,x2) = max{x1 + 5, x2 + 2}
        false_0_A = 6
        |app'|_A(x1,x2) = max{26, x1 - 29, x2 - 110}
        add_0_A = 1
        add_1_A(x1) = 25
        app_0_A = 28
        app_1_A(x1) = max{25, x1 - 111}
        filter_0_A = 23
        filter_1_A(x1) = max{24, x1 - 110}
        filter2_0_A = 24
        filter2_1_A(x1) = max{25, x1 - 111}
        filter2_2_A(x1,x2) = max{26, x1 - 141, x2 - 110}
        high_0_A = 1
        high_1_A(x1) = max{2, x1 - 111}
        if_high_0_A = 1
        if_high_1_A(x1) = max{26, x1 - 111}
        if_high_2_A(x1,x2) = max{26, x1 - 140, x2 - 110}
        if_low_0_A = 24
        if_low_1_A(x1) = max{25, x1 - 111}
        if_low_2_A(x1,x2) = max{26, x1 - 140, x2 - 110}
        le_0_A = 1
        le_1_A(x1) = max{2, x1 - 111}
        quot_0_A = 1
        quot_1_A(x1) = max{25, x1 - 110}
        quot_2_A(x1,x2) = max{24, x1 - 140, x2 - 119}
        |0|_0_A = 23
        s_1_A(x1) = x1 + 22
        true_0_A = 3
        low_2_A(x1,x2) = max{50, x2 + 1}
        nil_0_A = 0
        high_2_A(x1,x2) = max{x1 + 67, x2}
        quicksort_1_A(x1) = max{37, x1 - 95}
        filter_2_A(x1,x2) = max{25, x1 - 230, x2 - 348}
  
    base order 2:
  
      precedence:
      
        nil_0 = filter_2 > low_2# = add_2 = |app'| = low_2 = high_2 > if_low_3# = filter_1 = quot_0 = quot_2 = |0|_0 = s_1 = quicksort_1 > false_0 = app_1 = quot_1 > le_2 = add_1 = app_0 = filter_0 = filter2_2 = high_1 = if_high_0 = if_high_2 = if_low_2 = le_0 > add_0 = filter2_1 = high_0 = if_high_1 = if_low_1 = le_1 = true_0 > filter2_0 = if_low_0
      
  
    partial status:
  
      pi(low_2#) = []
      pi(add_2) = [2]
      pi(if_low_3#) = []
      pi(le_2) = [1, 2]
      pi(false_0) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(true_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = [1, 2]
      pi(quicksort_1) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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 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

  r1, r2, r3, r5, r7, r8, r9, r11, r13, r14, r17, r19, r21, r23, r25, r28, r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le_2#_A(x1,x2) = max{1, x1 - 10, x2 - 13}
        s_1_A(x1) = max{12, x1 + 9}
        |app'|_A(x1,x2) = max{69, x1 - 71, x2 - 15}
        add_0_A = 70
        add_1_A(x1) = max{15, x1 - 16}
        add_2_A(x1,x2) = max{14, x1 - 246, x2 - 277}
        app_0_A = 1
        app_1_A(x1) = max{2, x1 - 16}
        filter_0_A = 70
        filter_1_A(x1) = max{48, x1 - 230}
        filter2_0_A = 70
        filter2_1_A(x1) = max{1, x1 - 15}
        filter2_2_A(x1,x2) = max{69, x1 - 86, x2 - 15}
        filter2_3_A(x1,x2,x3) = max{49, x1 - 158, x2 - 157, x3 - 157}
        high_0_A = 1
        high_1_A(x1) = max{11, x1 - 283}
        if_high_0_A = 10
        if_high_1_A(x1) = max{11, x1 - 147}
        if_high_2_A(x1,x2) = max{12, x1 - 218, x2 - 195}
        if_low_1_A(x1) = max{70, x1 - 16}
        if_low_2_A(x1,x2) = max{67, x1 - 88, x2 - 16}
        le_0_A = 1
        le_1_A(x1) = max{0, x1 - 16}
        low_0_A = 12
        low_1_A(x1) = max{13, x1 - 16}
        map_0_A = 68
        map_1_A(x1) = max{69, x1 - 246}
        minus_0_A = 2
        minus_1_A(x1) = max{3, x1 - 16}
        quot_0_A = 11
        quot_1_A(x1) = max{12, x1 - 16}
        quot_2_A(x1,x2) = max{13, x1 - 88, x2 - 92}
        |0|_0_A = 11
        le_2_A(x1,x2) = max{11, x2 - 283}
        true_0_A = 1
        false_0_A = 10
        low_2_A(x1,x2) = max{14, x1 - 282, x2 - 556}
        nil_0_A = 11
        high_2_A(x1,x2) = max{12, x1 - 282, x2 - 558}
        quicksort_1_A(x1) = max{68, x1 - 35}
        filter_2_A(x1,x2) = max{49, x1 - 302, x2 - 230}
  
    base order 2:
  
      precedence:
      
        filter_1 > nil_0 = filter_2 > |app'| > le_2# = app_0 = filter_0 = filter2_3 = quot_2 > add_2 = filter2_1 = filter2_2 = map_1 = quot_1 > add_1 = filter2_0 = if_high_2 = if_low_2 = low_1 = map_0 = minus_1 = quot_0 = quicksort_1 > add_0 = app_1 = if_high_1 = if_low_1 = low_0 = minus_0 = low_2 = high_2 > s_1 = high_1 = if_high_0 = le_1 = le_2 > high_0 = le_0 = |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = [1]
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(add_2) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_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: 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

  r1, r2, r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r31, r32, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quot_2#_A(x1,x2) = max{1, x1 - 15}
        s_1_A(x1) = x1 + 19
        minus_2_A(x1,x2) = max{14, x1 + 3}
        |app'|_A(x1,x2) = max{12, x1 - 14, x2 - 8}
        add_0_A = 1
        add_1_A(x1) = max{2, x1 - 9}
        add_2_A(x1,x2) = max{7, x1 - 69, x2 - 50}
        app_0_A = 27
        app_1_A(x1) = max{13, x1 - 8}
        filter_0_A = 8
        filter_1_A(x1) = max{9, x1 - 54}
        filter2_0_A = 1
        filter2_1_A(x1) = max{11, x1 - 9}
        filter2_2_A(x1,x2) = max{1, x1 - 24, x2 - 9}
        filter2_3_A(x1,x2,x3) = max{10, x1 - 39, x2 - 24, x3 - 9}
        high_0_A = 1
        high_1_A(x1) = max{2, x1 - 44}
        if_high_0_A = 1
        if_high_1_A(x1) = max{2, x1 - 9}
        if_high_2_A(x1,x2) = max{3, x1 - 24, x2 - 9}
        if_low_0_A = 2
        if_low_1_A(x1) = max{3, x1 - 9}
        if_low_2_A(x1,x2) = max{4, x1 - 23, x2 - 8}
        le_0_A = 13
        le_1_A(x1) = max{1, x1 - 55}
        low_0_A = 1
        low_1_A(x1) = max{2, x1 - 69}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 84}
        minus_0_A = 1
        minus_1_A(x1) = max{0, x1 - 9}
        |0|_0_A = 18
        le_2_A(x1,x2) = max{6, x1 - 70, x2 - 20}
        true_0_A = 1
        false_0_A = 2
        low_2_A(x1,x2) = max{5, x1 - 68, x2 - 49}
        nil_0_A = 0
        high_2_A(x1,x2) = max{3, x1 - 59, x2 - 59}
        quicksort_1_A(x1) = max{2, x1 - 10}
        filter_2_A(x1,x2) = max{10, x1 - 69, x2 - 50}
  
    base order 2:
  
      precedence:
      
        low_2 > minus_2 = le_2 > s_1 > quot_2# > |app'| > filter_0 = filter2_3 = |0|_0 = filter_2 > filter_1 = filter2_2 = false_0 > filter2_1 = map_0 = high_2 > add_0 = add_2 = filter2_0 = high_1 = if_high_0 = if_high_2 = if_low_2 = le_1 = low_1 = map_1 = minus_1 > add_1 = high_0 = if_high_1 = if_low_1 = le_0 = low_0 = minus_0 = true_0 = quicksort_1 > if_low_0 = nil_0 > app_1 > app_0
      
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = [1]
      pi(minus_2) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(add_2) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_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: 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

  r1, r2, r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r33, r35, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_2#_A(x1,x2) = max{33, x1 + 29, x2 - 33}
        s_1_A(x1) = max{32, x1 + 5}
        |app'|_A(x1,x2) = max{31, x1 - 32, x2 - 15}
        add_0_A = 1
        add_1_A(x1) = max{15, x1 - 16}
        add_2_A(x1,x2) = max{14, x1 - 130, x2 - 141}
        app_0_A = 10
        app_1_A(x1) = max{11, x1 - 15}
        filter_0_A = 27
        filter_1_A(x1) = max{28, x1 - 114}
        filter2_0_A = 29
        filter2_1_A(x1) = max{30, x1 - 16}
        filter2_2_A(x1,x2) = max{31, x1 - 48, x2 - 15}
        filter2_3_A(x1,x2,x3) = max{31, x1 - 81, x2 - 48, x3 - 15}
        high_0_A = 23
        high_1_A(x1) = max{24, x1 - 118}
        if_high_0_A = 20
        if_high_1_A(x1) = max{21, x1 - 16}
        if_high_2_A(x1,x2) = max{22, x1 - 49, x2 - 16}
        if_low_0_A = 1
        if_low_1_A(x1) = max{11, x1 - 15}
        if_low_2_A(x1,x2) = max{12, x1 - 48, x2 - 16}
        le_0_A = 1
        le_1_A(x1) = max{2, x1 - 16}
        low_0_A = 1
        low_1_A(x1) = max{2, x1 - 131}
        map_0_A = 1
        map_1_A(x1) = max{29, x1 - 80}
        minus_0_A = 1
        minus_1_A(x1) = max{2, x1 - 16}
        quot_0_A = 1
        quot_1_A(x1) = max{2, x1 - 16}
        quot_2_A(x1,x2) = max{31, x1 - 48, x2 - 49}
        |0|_0_A = 30
        le_2_A(x1,x2) = max{9, x2 - 131}
        true_0_A = 1
        false_0_A = 6
        low_2_A(x1,x2) = max{10, x1 - 130, x2 - 158}
        nil_0_A = 0
        high_2_A(x1,x2) = max{23, x1 - 117, x2 - 157}
        quicksort_1_A(x1) = max{13, x1 - 50}
        filter_2_A(x1,x2) = max{29, x1 - 147, x2 - 114}
  
    base order 2:
  
      precedence:
      
        nil_0 = filter_2 > |app'| = filter_0 > app_0 = filter_1 = filter2_3 = map_0 = quot_0 = quot_2 > filter2_2 = if_high_2 = map_1 = quot_1 = high_2 > add_2 = filter2_1 = high_1 = if_high_1 = if_low_2 = minus_1 > add_1 = filter2_0 = high_0 = if_high_0 = if_low_1 = minus_0 = low_2 > minus_2# = s_1 = add_0 = if_low_0 = le_1 = low_1 = le_2 = quicksort_1 > le_0 = low_0 = |0|_0 > app_1 = true_0 = false_0
      
  
    partial status:
  
      pi(minus_2#) = [1]
      pi(s_1) = [1]
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(add_2) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(low_0) = []
      pi(low_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.