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, r4, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r38, r39, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x1 + x2 + 12
          add_2_A(x1,x2) = x1 + x2 + 11
          |app'|#_A(x1,x2) = x1 + x2 + 12
          map_1_A(x1) = x1 + 13
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 30
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 26
          false_0_A() = 17
          filter_2#_A(x1,x2) = x1 + x2 + 16
          filter_1_A(x1) = x1 + 5
          |app'|_A(x1,x2) = x1 + x2 + 26
          true_0_A() = 1
          add_0_A() = 1
          add_1_A(x1) = x1 + 2
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          app_2_A(x1,x2) = x1 + x2 + 12
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 5
          high_0_A() = 1
          high_1_A(x1) = x1 + 2
          if_high_0_A() = 1
          if_high_1_A(x1) = x1 + 2
          if_high_2_A(x1,x2) = x1 + x2 + 3
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 20
          low_0_A() = 1
          low_1_A(x1) = x1 + 2
          map_0_A() = 1
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + x2 + 3
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 18
          s_0_A() = 20
          s_1_A(x1) = x1 + 19
          |0|_0_A() = 18
          quot_2_A(x1,x2) = x1 + 17
          nil_0_A() = 1
          low_2_A(x1,x2) = x1 + x2 + 18
          high_2_A(x1,x2) = x1 + x2 + 10
          quicksort_1_A(x1) = x1 + 8
          filter_2_A(x1,x2) = x1 + x2 + 18
  
    base order 2:
  
      precedence:
      
        app_2 = filter_2 > |app'|# = filter_1 = quicksort_1 > map_2# = filter_2# = low_2 > add_2 = |app'| = filter_0 = high_1 = high_2 > map_1 = filter2_4# = add_1 = app_1 = high_0 = if_low_2 = le_2 = low_1 = minus_2 = quot_1 = nil_0 > filter2_3 = false_0 = add_0 = app_0 = filter2_2 = if_high_2 = if_low_1 = le_1 = low_0 = map_0 = minus_1 = quot_0 = quot_2 > filter2_1 = if_high_1 = if_low_0 = le_0 = minus_0 = s_1 > filter2_0 = if_high_0 = s_0 = |0|_0 > true_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(add_2) = [2]
      pi(|app'|#) = [2]
      pi(map_1) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(|app'|) = []
      pi(true_0) = []
      pi(add_0) = []
      pi(add_1) = [1]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = [1, 2]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = []
      pi(if_high_2) = [1, 2]
      pi(if_low_0) = []
      pi(if_low_1) = [1]
      pi(if_low_2) = [2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = [1]
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = [1, 2]
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(nil_0) = []
      pi(low_2) = [2]
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_2) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: |app'|#(map_1(x1),y) -> map_2#(x1,y)
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:

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


-- Reduction pair.

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

p1: filter_2#(f,add_2(x,xs)) -> |app'|#(f,x)
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#(true_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,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, r4, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r38, r39, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 11
          add_2_A(x1,x2) = x1 + x2 + 8
          |app'|#_A(x1,x2) = x1 + x2 + 10
          filter_1_A(x1) = x1 + 2
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 12
          |app'|_A(x1,x2) = x1 + x2 + 18
          true_0_A() = 1
          false_0_A() = 1
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          add_0_A() = 1
          add_1_A(x1) = x1 + 2
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          app_2_A(x1,x2) = x1 + x2 + 3
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          high_0_A() = 1
          high_1_A(x1) = x1 + 2
          if_high_0_A() = 2
          if_high_1_A(x1) = x1 + 1
          if_high_2_A(x1,x2) = x1 + x2 + 2
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 9
          low_0_A() = 1
          low_1_A(x1) = x1 + 6
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + x2 + 5
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          |0|_0_A() = 2
          quot_2_A(x1,x2) = x1 + 1
          nil_0_A() = 1
          low_2_A(x1,x2) = x1 + 5
          high_2_A(x1,x2) = x1 + x2 + 4
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 11
  
    base order 2:
  
      precedence:
      
        |app'|# > filter2_4# > filter_2# = filter_2 > |app'| > filter_1 = app_2 > add_2 = app_1 = filter_0 = low_1 = minus_2 > filter2_3 = add_1 = app_0 = low_0 = map_1 = minus_1 = quicksort_1 > add_0 = filter2_2 = if_high_0 = map_0 = minus_0 = high_2 > filter2_1 = high_1 = if_high_1 = le_2 > filter2_0 = high_0 = if_high_2 = le_1 = s_1 = quot_2 = low_2 > true_0 = le_0 = quot_1 = s_0 = nil_0 > if_low_2 = quot_0 = |0|_0 > false_0 = if_low_1 > if_low_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(add_2) = [2]
      pi(|app'|#) = [2]
      pi(filter_1) = [1]
      pi(filter2_4#) = []
      pi(|app'|) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(filter2_3) = [3]
      pi(add_0) = []
      pi(add_1) = [1]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = [1]
      pi(if_low_0) = []
      pi(if_low_1) = [1]
      pi(if_low_2) = [1, 2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(quot_2) = [1]
      pi(nil_0) = []
      pi(low_2) = []
      pi(high_2) = []
      pi(quicksort_1) = []
      pi(filter_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

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

  {p2, p3, p4}


-- Reduction pair.

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

p1: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p3: 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, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x2
          add_2_A(x1,x2) = x1 + x2 + 8
          filter2_4#_A(x1,x2,x3,x4) = x4 + 7
          |app'|_A(x1,x2) = x1 + x2 + 2
          false_0_A() = 1
          true_0_A() = 6
          add_0_A() = 6
          add_1_A(x1) = x1 + 7
          app_0_A() = 7
          app_1_A(x1) = x1 + 8
          filter_0_A() = 5
          filter_1_A(x1) = x1 + 6
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          high_0_A() = 1
          high_1_A(x1) = x1 + 2
          if_high_0_A() = 1
          if_high_1_A(x1) = x1 + 2
          if_high_2_A(x1,x2) = x1 + x2 + 3
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x2 + 3
          le_0_A() = 7
          le_1_A(x1) = x1 + 8
          le_2_A(x1,x2) = x1 + x2 + 9
          low_0_A() = 7
          low_1_A(x1) = x1 + 8
          map_0_A() = 7
          map_1_A(x1) = x1 + 8
          minus_0_A() = 3
          minus_1_A(x1) = x1 + 4
          minus_2_A(x1,x2) = x1 + x2 + 5
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 2
          s_1_A(x1) = x1 + 3
          |0|_0_A() = 2
          quot_2_A(x1,x2) = x1 + 1
          high_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 8
          quicksort_1_A(x1) = x1 + 4
          filter_2_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      precedence:
      
        low_1 > add_1 = filter_2 > add_2 = add_0 = filter_1 = filter2_0 = low_0 = minus_0 = high_2 = quicksort_1 > |app'| = filter_0 = nil_0 > app_1 = filter2_3 = high_1 = if_high_2 = le_2 = map_1 = minus_2 = quot_1 > true_0 = app_0 = filter2_2 = high_0 = if_high_1 = if_low_2 = le_1 = map_0 = minus_1 = quot_0 = quot_2 > filter_2# = filter2_1 = if_high_0 = if_low_1 = le_0 = s_1 > filter2_4# = false_0 = if_low_0 = s_0 > |0|_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(add_2) = [1]
      pi(filter2_4#) = []
      pi(|app'|) = [2]
      pi(false_0) = []
      pi(true_0) = []
      pi(add_0) = []
      pi(add_1) = [1]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [3]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = [1, 2]
      pi(if_low_0) = []
      pi(if_low_1) = [1]
      pi(if_low_2) = [2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = [1]
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = [1]
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(high_2) = []
      pi(nil_0) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = [1]

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: filter_2#(f,add_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,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:

  {p1, p2}


-- Reduction pair.

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

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

  r1, r2, r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x2
          add_2_A(x1,x2) = x1 + x2 + 7
          filter2_4#_A(x1,x2,x3,x4) = x4 + 1
          |app'|_A(x1,x2) = x1 + x2 + 8
          false_0_A() = 1
          add_0_A() = 1
          add_1_A(x1) = x1 + 2
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + 4
          high_0_A() = 1
          high_1_A(x1) = x1 + 2
          if_high_0_A() = 2
          if_high_1_A(x1) = x1 + 1
          if_high_2_A(x1,x2) = x1 + x2 + 2
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 8
          low_0_A() = 1
          low_1_A(x1) = x1 + 2
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 4
          minus_2_A(x1,x2) = x1 + x2 + 11
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 8
          s_1_A(x1) = x1 + 9
          |0|_0_A() = 2
          quot_2_A(x1,x2) = x1 + 1
          true_0_A() = 3
          low_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
          high_2_A(x1,x2) = x1 + x2 + 4
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        filter_2 > nil_0 > low_2 > add_2 > if_high_1 > if_high_2 = minus_2 > |app'| = app_0 = filter_0 = if_high_0 > add_1 = app_1 = filter_1 = filter2_3 = low_1 = map_1 = minus_1 = quot_1 > add_0 = filter2_2 = if_low_2 = low_0 = map_0 = minus_0 = quot_0 = quot_2 = quicksort_1 > filter2_1 = high_1 = if_low_1 = high_2 > filter2_0 = high_0 = if_low_0 = le_2 > false_0 = le_1 > filter_2# = le_0 = s_1 > filter2_4# = s_0 = |0|_0 > true_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(add_2) = []
      pi(filter2_4#) = []
      pi(|app'|) = []
      pi(false_0) = []
      pi(add_0) = []
      pi(add_1) = [1]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(filter2_3) = [2]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = [2]
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = [2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = [1]
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(true_0) = []
      pi(low_2) = [1]
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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: 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

  r1, r2, r3, r4, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r38, r39, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x2
          add_2_A(x1,x2) = x1 + x2 + 8
          |app'|_A(x1,x2) = x1 + x2 + 18
          add_0_A() = 2
          add_1_A(x1) = x1 + 1
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          app_2_A(x1,x2) = x1 + x2 + 9
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          high_0_A() = 1
          high_1_A(x1) = x1 + 2
          if_high_0_A() = 1
          if_high_1_A(x1) = x1 + 2
          if_high_2_A(x1,x2) = x1 + x2 + 3
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 9
          low_0_A() = 1
          low_1_A(x1) = x1 + 2
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + x2 + 5
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          |0|_0_A() = 2
          quot_2_A(x1,x2) = x1 + 1
          true_0_A() = 1
          false_0_A() = 1
          nil_0_A() = 1
          low_2_A(x1,x2) = x1 + x2 + 3
          high_2_A(x1,x2) = x1 + x2 + 5
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 11
  
    base order 2:
  
      precedence:
      
        if_low_2 > if_low_1 = filter_2 > |app'| > minus_2 = quot_1 > quot_2 = high_2 > app_2 = filter_0 = filter2_3 = le_2 = s_1 > add_2 = add_0 = app_1 = filter_1 = filter2_2 = high_1 = if_low_0 = le_1 = minus_1 = quot_0 = s_0 = |0|_0 = low_2 > map_2# = add_1 = app_0 = filter2_1 = high_0 = le_0 = low_1 = map_1 = minus_0 = true_0 = false_0 = quicksort_1 > filter2_0 = if_high_2 = low_0 = map_0 = nil_0 > if_high_1 > if_high_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(add_2) = [1, 2]
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = [1]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(filter2_3) = [3]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = [2]
      pi(if_low_0) = []
      pi(if_low_1) = [1]
      pi(if_low_2) = [1]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = [1]
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(low_2) = [2]
      pi(high_2) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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, r7, r8, r9, r11, r12, r13, r14, r15, r16, r17, r19, r20, r21, r23, r25, r28, r30, r31, r32, 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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          quicksort_1#_A(x1) = x1
          add_2_A(x1,x2) = x1 + x2 + 8
          high_2_A(x1,x2) = x2 + 7
          low_2_A(x1,x2) = x1 + x2 + 2
          |app'|_A(x1,x2) = x2 + 16
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 2
          app_1_A(x1) = 1
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x2 + 2
          filter2_3_A(x1,x2,x3) = 1
          high_0_A() = 2
          high_1_A(x1) = 1
          if_high_0_A() = 1
          if_high_1_A(x1) = x1 + 9
          if_high_2_A(x1,x2) = 8
          if_high_3_A(x1,x2,x3) = x3 + 7
          if_low_0_A() = 3
          if_low_1_A(x1) = 2
          if_low_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          le_2_A(x1,x2) = x2 + 4
          low_0_A() = 2
          low_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 5
          minus_2_A(x1,x2) = x1 + x2 + 17
          |0|_0_A() = 7
          quot_2_A(x1,x2) = x1 + 17
          true_0_A() = 3
          false_0_A() = 6
          nil_0_A() = 1
          if_low_3_A(x1,x2,x3) = x2 + x3 + 2
          filter_2_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      precedence:
      
        quicksort_1# > high_2 = low_2 = add_0 = filter2_0 = if_high_1 = map_0 = quot_0 > if_high_0 = le_2 > s_0 = false_0 > s_1 = minus_2 = |0|_0 = quot_2 > |app'| = app_0 = high_0 = if_low_0 = le_0 = low_0 = minus_0 > app_1 = filter2_2 = high_1 = if_high_3 = if_low_1 = le_1 = low_1 = map_1 = minus_1 = quot_1 > add_2 = filter2_1 = if_high_2 = if_low_3 > add_1 = if_low_2 > filter2_3 = filter_2 > true_0 = nil_0
      
  
    partial status:
  
      pi(quicksort_1#) = [1]
      pi(add_2) = [1]
      pi(high_2) = []
      pi(low_2) = [2]
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [2]
      pi(filter2_3) = []
      pi(high_0) = []
      pi(high_1) = []
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = []
      pi(if_high_3) = []
      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(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_low_3) = [3]
      pi(filter_2) = [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, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r30, r31, r32, r33, r36, r38, r39, r40, r44, r45, r46, r47, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          quicksort_1#_A(x1) = x1
          add_2_A(x1,x2) = x1 + x2 + 10
          high_2_A(x1,x2) = x1 + x2 + 6
          |app'|_A(x1,x2) = x2 + 9
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 2
          app_1_A(x1) = 1
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          high_0_A() = 8
          high_1_A(x1) = 7
          if_high_0_A() = 9
          if_high_1_A(x1) = 8
          if_high_2_A(x1,x2) = 7
          if_low_0_A() = 1
          if_low_1_A(x1) = 2
          if_low_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          low_0_A() = 2
          low_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          minus_2_A(x1,x2) = x1 + x2 + 14
          |0|_0_A() = 2
          quot_2_A(x1,x2) = x1 + 10
          le_2_A(x1,x2) = x2 + 2
          false_0_A() = 1
          app_2_A(x1,x2) = x1 + x2 + 11
          nil_0_A() = 1
          low_2_A(x1,x2) = x1 + x2 + 4
          if_high_3_A(x1,x2,x3) = x2 + x3 + 6
          true_0_A() = 3
          filter_2_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        quicksort_1# > high_2 > app_2 = if_high_3 > add_2 = minus_2 = filter_2 > quot_2 > |0|_0 = nil_0 = low_2 > |app'| = add_0 = app_0 = filter2_0 = high_0 = if_high_0 = if_high_2 = le_0 = low_0 = map_0 = minus_0 = quot_0 = true_0 > add_1 = app_1 = filter_1 = filter2_1 = high_1 = if_high_1 = if_low_1 = low_1 = map_1 = minus_1 = quot_1 = s_1 = le_2 > filter_0 = filter2_2 = if_low_0 = if_low_2 = le_1 = s_0 = false_0 > filter2_3
      
  
    partial status:
  
      pi(quicksort_1#) = []
      pi(add_2) = []
      pi(high_2) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = [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) = [1]
      pi(minus_2) = [2]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(le_2) = []
      pi(false_0) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(low_2) = [2]
      pi(if_high_3) = []
      pi(true_0) = []
      pi(filter_2) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: app_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

  r3, r5, r7, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r31, r32, r33, r35, r36, r37, r40, r44, r48, r52, r53, r54, r55

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app_2#_A(x1,x2) = x1
          add_2_A(x1,x2) = x1 + x2 + 6
          |app'|_A(x1,x2) = 7
          app_0_A() = 2
          app_1_A(x1) = 1
          filter_0_A() = 7
          filter_1_A(x1) = 6
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + x2 + 2
          filter2_3_A(x1,x2,x3) = 1
          high_0_A() = 2
          high_1_A(x1) = 1
          if_high_0_A() = 3
          if_high_1_A(x1) = 2
          if_high_2_A(x1,x2) = 1
          if_low_0_A() = 3
          if_low_1_A(x1) = 2
          if_low_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          low_0_A() = 2
          low_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          minus_2_A(x1,x2) = x1 + x2 + 18
          |0|_0_A() = 2
          s_1_A(x1) = x1 + 9
          quot_2_A(x1,x2) = x1 + 8
          le_2_A(x1,x2) = x1 + x2 + 8
          true_0_A() = 1
          false_0_A() = 1
          low_2_A(x1,x2) = x2 + 3
          nil_0_A() = 1
          high_2_A(x1,x2) = x1 + x2 + 3
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 5
          filter2_4_A(x1,x2,x3,x4) = x2 + x3 + x4 + 11
  
    base order 2:
  
      precedence:
      
        if_low_0 > if_low_1 = minus_2 = filter_2 > filter2_2 = if_low_2 = quot_2 = filter2_4 > add_2 = |app'| = app_0 = filter_0 = filter2_0 = high_0 = if_high_0 = le_0 = low_0 = map_0 = minus_0 = quot_0 = s_1 = le_2 = high_2 > app_2# = filter_1 = filter2_1 = filter2_3 = high_1 = if_high_1 = le_1 = map_1 = minus_1 = quot_1 = |0|_0 > app_1 = if_high_2 = true_0 = quicksort_1 > low_2 > low_1 = false_0 = nil_0
      
  
    partial status:
  
      pi(app_2#) = []
      pi(add_2) = []
      pi(|app'|) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [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(minus_2) = [1]
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(quot_2) = []
      pi(le_2) = [1]
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = [2]
      pi(quicksort_1) = [1]
      pi(filter_2) = []
      pi(filter2_4) = [4]

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

  r1, r3, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r30, r31, r32, r33, r36, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          high_2#_A(x1,x2) = x2 + 1
          add_2_A(x1,x2) = x1 + x2 + 27
          if_high_3#_A(x1,x2,x3) = x3
          le_2_A(x1,x2) = x1 + 1
          false_0_A() = 2
          true_0_A() = 5
          |app'|_A(x1,x2) = x2 + 11
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 1
          app_1_A(x1) = 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 11
          filter2_1_A(x1) = 10
          filter2_2_A(x1,x2) = 9
          filter2_3_A(x1,x2,x3) = 8
          high_0_A() = 2
          high_1_A(x1) = 1
          if_high_0_A() = 1
          if_high_1_A(x1) = 2
          if_high_2_A(x1,x2) = 1
          if_low_0_A() = 4
          if_low_1_A(x1) = 3
          if_low_2_A(x1,x2) = 2
          le_0_A() = 3
          le_1_A(x1) = 2
          low_0_A() = 2
          low_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 4
          minus_2_A(x1,x2) = x1 + x2 + 3
          |0|_0_A() = 3
          quot_2_A(x1,x2) = x1 + 12
          low_2_A(x1,x2) = x2 + 12
          nil_0_A() = 1
          high_2_A(x1,x2) = 12
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        high_2# = add_2 = filter_0 = high_0 = if_high_0 = if_low_0 = low_0 = map_0 = minus_2 = nil_0 = high_2 > if_high_3# = quot_2 = quicksort_1 > le_0 = |0|_0 = low_2 > false_0 = le_1 = s_1 > |app'| = add_0 = app_1 = filter2_0 = if_low_1 = if_low_2 = minus_0 = quot_0 = s_0 > add_1 = app_0 = filter2_1 = high_1 = if_high_1 = low_1 = minus_1 = quot_1 = filter_2 > le_2 = true_0 = filter_1 = filter2_2 = if_high_2 = map_1 > filter2_3
      
  
    partial status:
  
      pi(high_2#) = [2]
      pi(add_2) = [2]
      pi(if_high_3#) = []
      pi(le_2) = [1]
      pi(false_0) = []
      pi(true_0) = []
      pi(|app'|) = [2]
      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) = [1]
      pi(minus_2) = [2]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p2



-- SCC decomposition.

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#(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:

  {p1, p2}


-- 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#(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

  r1, r3, r7, r8, r11, r13, r14, r16, r17, r19, r21, r23, r25, r28, r31, r32, r33, r35, r36, r37, r38, r39, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          high_2#_A(x1,x2) = x2 + 4
          add_2_A(x1,x2) = x1 + x2 + 9
          if_high_3#_A(x1,x2,x3) = x3
          le_2_A(x1,x2) = x1 + x2 + 12
          true_0_A() = 5
          |app'|_A(x1,x2) = 10
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 10
          app_1_A(x1) = 9
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          high_0_A() = 2
          high_1_A(x1) = 1
          if_high_0_A() = 3
          if_high_1_A(x1) = 2
          if_high_2_A(x1,x2) = 1
          if_low_0_A() = 3
          if_low_1_A(x1) = 2
          if_low_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          low_0_A() = 2
          low_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          minus_2_A(x1,x2) = x1 + x2 + 23
          |0|_0_A() = 6
          s_1_A(x1) = x1 + 11
          quot_2_A(x1,x2) = x1 + 11
          false_0_A() = 3
          app_2_A(x1,x2) = x1 + x2 + 8
          nil_0_A() = 1
          low_2_A(x1,x2) = x1 + x2 + 4
          high_2_A(x1,x2) = x1 + x2 + 6
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 12
  
    base order 2:
  
      precedence:
      
        minus_2 = app_2 = low_2 = filter_2 > |app'| = add_0 = app_0 = filter2_0 = if_low_0 = low_0 = map_0 = minus_0 > add_2 = add_1 = app_1 = filter2_1 = high_0 = if_low_1 = le_0 = low_1 = map_1 = minus_1 = quot_0 > if_high_0 = if_high_2 = if_low_2 = high_2 > true_0 = high_1 = if_high_1 = quicksort_1 > filter2_2 = |0|_0 = nil_0 > false_0 > le_2 > high_2# = le_1 = s_1 = quot_2 > if_high_3# = quot_1
      
  
    partial status:
  
      pi(high_2#) = []
      pi(add_2) = []
      pi(if_high_3#) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(|app'|) = []
      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(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(minus_2) = [2]
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(quot_2) = []
      pi(false_0) = []
      pi(app_2) = [1, 2]
      pi(nil_0) = []
      pi(low_2) = []
      pi(high_2) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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))

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, r31, r32, r33, r35, r36, r37, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_low_3#_A(x1,x2,x3) = x3
          true_0_A() = 1
          add_2_A(x1,x2) = x1 + x2 + 9
          low_2#_A(x1,x2) = x2 + 2
          le_2_A(x1,x2) = x1 + x2 + 14
          false_0_A() = 3
          |app'|_A(x1,x2) = x2 + 10
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 2
          app_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          high_0_A() = 2
          high_1_A(x1) = 1
          if_high_0_A() = 1
          if_high_1_A(x1) = 2
          if_high_2_A(x1,x2) = 3
          if_low_0_A() = 4
          if_low_1_A(x1) = 3
          if_low_2_A(x1,x2) = 2
          le_0_A() = 2
          le_1_A(x1) = 1
          low_0_A() = 2
          low_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 5
          minus_2_A(x1,x2) = x1 + x2 + 17
          |0|_0_A() = 4
          quot_2_A(x1,x2) = x1 + 11
          low_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
          high_2_A(x1,x2) = x1 + x2 + 6
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 11
  
    base order 2:
  
      precedence:
      
        s_1 = quot_2 > minus_2 = |0|_0 = filter_2 > true_0 = false_0 > le_2 = high_2 > low_2# = |app'| = add_0 = filter_0 = filter2_0 = high_0 = if_low_0 = le_0 = low_0 = map_0 = minus_0 = quot_0 = s_0 > add_2 = app_1 = filter_1 = filter2_1 = high_1 = if_high_2 = le_1 = map_1 = minus_1 = quot_1 = nil_0 > if_low_3# = add_1 = app_0 = filter2_2 = if_high_1 = if_low_1 = if_low_2 = quicksort_1 > filter2_3 = if_high_0 = low_2 > low_1
      
  
    partial status:
  
      pi(if_low_3#) = []
      pi(true_0) = []
      pi(add_2) = [2]
      pi(low_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) = [1]
      pi(minus_2) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(low_2) = []
      pi(nil_0) = []
      pi(high_2) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p3



-- SCC decomposition.

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))

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: 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))

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, r7, r8, r11, r13, r16, r17, r19, r21, r23, r25, r28, r31, r32, r33, r35, r36, r37, r38, r39, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_low_3#_A(x1,x2,x3) = x3
          true_0_A() = 6
          add_2_A(x1,x2) = x1 + x2 + 12
          low_2#_A(x1,x2) = x2 + 3
          le_2_A(x1,x2) = x1 + x2 + 16
          |app'|_A(x1,x2) = 3
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 3
          app_1_A(x1) = 2
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          high_0_A() = 2
          high_1_A(x1) = 1
          if_high_0_A() = 2
          if_high_1_A(x1) = 1
          if_low_0_A() = 3
          if_low_1_A(x1) = 2
          if_low_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          low_0_A() = 2
          low_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          minus_2_A(x1,x2) = x1 + x2 + 9
          |0|_0_A() = 2
          s_1_A(x1) = x1 + 4
          quot_2_A(x1,x2) = x1 + 4
          false_0_A() = 1
          app_2_A(x1,x2) = x1 + x2 + 13
          nil_0_A() = 1
          low_2_A(x1,x2) = x1 + x2 + 5
          high_2_A(x1,x2) = x1 + x2 + 7
          quicksort_1_A(x1) = x1 + 4
          filter_2_A(x1,x2) = x1 + x2 + 14
  
    base order 2:
  
      precedence:
      
        app_2 = low_2 > add_2 = add_0 = app_0 = if_low_0 = le_0 = quot_0 = minus_2 = high_2 = quicksort_1 = filter_2 > le_2 = nil_0 > true_0 > low_2# = |0|_0 > if_low_3# = false_0 > s_1 = quot_2 > |app'| = filter2_0 = high_0 = if_high_0 = low_0 = map_0 = minus_0 > add_1 = app_1 = filter2_1 = high_1 = if_high_1 = if_low_1 = le_1 = low_1 = map_1 = minus_1 = quot_1 > filter2_2 = if_low_2
      
  
    partial status:
  
      pi(if_low_3#) = []
      pi(true_0) = []
      pi(add_2) = [1]
      pi(low_2#) = []
      pi(le_2) = []
      pi(|app'|) = []
      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(minus_2) = [2]
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(quot_2) = []
      pi(false_0) = []
      pi(app_2) = [1, 2]
      pi(nil_0) = []
      pi(low_2) = []
      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))

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, r3, r4, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r38, r39, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          le_2#_A(x1,x2) = x1 + x2 + 4
          s_1_A(x1) = x1 + 3
          |app'|_A(x1,x2) = x1 + x2 + 22
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          app_2_A(x1,x2) = x1 + x2 + 11
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 14
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          high_0_A() = 1
          high_1_A(x1) = x1 + 2
          if_high_0_A() = 1
          if_high_1_A(x1) = x1 + 2
          if_high_2_A(x1,x2) = x1 + x2 + 3
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 16
          low_0_A() = 1
          low_1_A(x1) = x1 + 2
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + x2 + 5
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 1
          |0|_0_A() = 2
          quot_2_A(x1,x2) = x1 + 1
          true_0_A() = 1
          false_0_A() = 1
          nil_0_A() = 1
          add_2_A(x1,x2) = x1 + x2 + 10
          low_2_A(x1,x2) = x1 + x2 + 5
          high_2_A(x1,x2) = x1 + x2 + 7
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        filter_2 > |app'| > le_2# = minus_2 > s_1 = quot_2 = high_2 > app_2 = filter2_3 = le_2 > add_0 = app_1 = filter_1 = filter2_0 = filter2_2 = high_1 = le_1 = minus_1 = quot_1 = s_0 = add_2 > add_1 = app_0 = filter_0 = filter2_1 = high_0 = le_0 = map_1 = minus_0 = quot_0 = quicksort_1 > if_high_2 = map_0 = |0|_0 = low_2 > if_high_1 = if_low_2 = low_1 = true_0 = false_0 = nil_0 > if_high_0 = if_low_1 = low_0 > if_low_0
      
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [3]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = [1, 2]
      pi(if_low_0) = []
      pi(if_low_1) = [1]
      pi(if_low_2) = [2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(|0|_0) = []
      pi(quot_2) = [1]
      pi(true_0) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(add_2) = [1]
      pi(low_2) = []
      pi(high_2) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          quot_2#_A(x1,x2) = x1 + 1
          s_1_A(x1) = x1 + 4
          minus_2_A(x1,x2) = x1 + 2
          |app'|_A(x1,x2) = x1 + x2 + 13
          add_0_A() = 1
          add_1_A(x1) = x1 + 2
          add_2_A(x1,x2) = x1 + x2 + 8
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          high_0_A() = 1
          high_1_A(x1) = x1 + 5
          if_high_0_A() = 1
          if_high_1_A(x1) = x1 + 2
          if_high_2_A(x1,x2) = x1 + x2 + 1
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + 1
          low_0_A() = 1
          low_1_A(x1) = x1 + 6
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 3
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 6
          s_0_A() = 1
          |0|_0_A() = 2
          quot_2_A(x1,x2) = 5
          true_0_A() = 1
          false_0_A() = 3
          low_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 1
          high_2_A(x1,x2) = x1 + x2 + 4
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        |app'| = if_high_2 > add_2 = filter2_3 = if_high_1 = le_1 = map_1 = minus_1 = quot_1 = filter_2 > add_1 = filter2_2 = if_high_0 = if_low_2 = le_0 = map_0 = minus_0 = quot_0 = high_2 > add_0 = filter2_1 = high_1 = if_low_1 = nil_0 > filter2_0 = high_0 = if_low_0 = quicksort_1 > app_1 = low_2 > app_0 = low_1 = false_0 > low_0 = |0|_0 > quot_2# = s_1 = minus_2 = le_2 = quot_2 > s_0 = true_0
      
  
    partial status:
  
      pi(quot_2#) = [1]
      pi(s_1) = []
      pi(minus_2) = []
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = [1]
      pi(add_2) = [1, 2]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(filter2_3) = [3]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = []
      pi(if_low_0) = []
      pi(if_low_1) = []
      pi(if_low_2) = [2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(low_2) = [2]
      pi(nil_0) = []
      pi(high_2) = [2]
      pi(quicksort_1) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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, r3, r4, r5, r7, r8, r9, r11, r13, r14, r16, r17, r19, r20, r21, r23, r25, r26, r28, r30, r31, r32, r33, r35, r36, r37, r38, r39, r40, r44, r48, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_2#_A(x1,x2) = x1 + x2 + 4
          s_1_A(x1) = x1 + 3
          |app'|_A(x1,x2) = x1 + x2 + 22
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          app_2_A(x1,x2) = x1 + x2 + 11
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 14
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          high_0_A() = 1
          high_1_A(x1) = x1 + 2
          if_high_0_A() = 1
          if_high_1_A(x1) = x1 + 2
          if_high_2_A(x1,x2) = x1 + x2 + 3
          if_low_0_A() = 1
          if_low_1_A(x1) = x1 + 2
          if_low_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 16
          low_0_A() = 1
          low_1_A(x1) = x1 + 2
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + x2 + 5
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 1
          |0|_0_A() = 2
          quot_2_A(x1,x2) = x1 + 1
          true_0_A() = 1
          false_0_A() = 1
          nil_0_A() = 1
          add_2_A(x1,x2) = x1 + x2 + 10
          low_2_A(x1,x2) = x1 + x2 + 5
          high_2_A(x1,x2) = x1 + x2 + 7
          quicksort_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        add_0 > add_1 = filter_2 > |app'| > minus_2# = minus_2 > s_1 = quot_2 = high_2 > app_2 = le_2 > app_1 = filter_1 = filter2_3 = high_1 = le_1 = minus_1 = quot_1 = s_0 = add_2 > app_0 = filter_0 = filter2_0 = filter2_2 = high_0 = le_0 = map_1 = minus_0 = quot_0 = quicksort_1 > filter2_1 = if_high_2 = map_0 = |0|_0 = low_2 > if_high_1 = if_low_2 = low_1 = true_0 = false_0 = nil_0 > if_high_0 = if_low_1 = low_0 > if_low_0
      
  
    partial status:
  
      pi(minus_2#) = []
      pi(s_1) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [3]
      pi(high_0) = []
      pi(high_1) = [1]
      pi(if_high_0) = []
      pi(if_high_1) = [1]
      pi(if_high_2) = [1, 2]
      pi(if_low_0) = []
      pi(if_low_1) = [1]
      pi(if_low_2) = [2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(low_0) = []
      pi(low_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(|0|_0) = []
      pi(quot_2) = [1]
      pi(true_0) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(add_2) = [1]
      pi(low_2) = []
      pi(high_2) = []
      pi(quicksort_1) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.