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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x1 + x2
              add_2_A(x1,x2) = x1 + x2 + 12
              |app'|#_A(x1,x2) = x1 + x2
              map_1_A(x1) = x1 + 2
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
              filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 2
              false_0_A() = 8
              filter_2#_A(x1,x2) = x1 + x2 + 2
              filter_1_A(x1) = x1 + 2
              |app'|_A(x1,x2) = x2 + 1
              true_0_A() = 6
              map_2#!_A(x1,x2) = 0
              add_2!_A(x1,x2) = x1 + x2 + 11
              |app'|#!_A(x1,x2) = 3
              map_1!_A(x1) = x1 + 22
              filter2_3!_A(x1,x2,x3) = 2
              filter2_4#!_A(x1,x2,x3,x4) = 2
              false_0!_A() = 0
              filter_2#!_A(x1,x2) = 1
              filter_1!_A(x1) = x1 + 2
              |app'|!_A(x1,x2) = 21
              true_0!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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 + 6
              |app'|#_A(x1,x2) = x2
              map_1_A(x1) = 3
              filter2_3_A(x1,x2,x3) = x1
              filter2_4#_A(x1,x2,x3,x4) = x4
              false_0_A() = 16
              filter_2#_A(x1,x2) = x2
              filter_1_A(x1) = x1 + 12
              |app'|_A(x1,x2) = x1 + 1
              true_0_A() = 18
              map_2#!_A(x1,x2) = 0
              add_2!_A(x1,x2) = x2 + 39
              |app'|#!_A(x1,x2) = x1 + x2 + 38
              map_1!_A(x1) = 1
              filter2_3!_A(x1,x2,x3) = x3 + 5
              filter2_4#!_A(x1,x2,x3,x4) = x2 + x3 + x4 + 50
              false_0!_A() = 0
              filter_2#!_A(x1,x2) = 49
              filter_1!_A(x1) = x1
              |app'|!_A(x1,x2) = 7
              true_0!_A() = 48
        
  
    partial status:
  
      pi(map_2#) = [2]
      pi(add_2) = [1, 2]
      pi(|app'|#) = [2]
      pi(map_1) = []
      pi(filter2_3) = [1, 2, 3]
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = []
      pi(|app'|) = []
      pi(true_0) = []

The following rules are removed:

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

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              quicksort_1#_A(x1) = 0
              add_2_A(x1,x2) = x1 + x2 + 14
              high_2_A(x1,x2) = x2 + 7
              low_2_A(x1,x2) = x1 + x2 + 14
              le_2_A(x1,x2) = x1 + x2 + 3
              |0|_0_A() = 13
              true_0_A() = 1
              s_1_A(x1) = x1 + 21
              false_0_A() = 1
              nil_0_A() = 4
              if_low_3_A(x1,x2,x3) = x2 + x3 + 14
              if_high_3_A(x1,x2,x3) = x3 + 7
              quicksort_1#!_A(x1) = x1
              add_2!_A(x1,x2) = x2 + 20
              high_2!_A(x1,x2) = x1 + x2 + 15
              low_2!_A(x1,x2) = x1 + x2 + 21
              le_2!_A(x1,x2) = x1 + 2
              |0|_0!_A() = 13
              true_0!_A() = 14
              s_1!_A(x1) = 23
              false_0!_A() = 22
              nil_0!_A() = 0
              if_low_3!_A(x1,x2,x3) = x2 + x3 + 20
              if_high_3!_A(x1,x2,x3) = x2 + x3 + 14
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              quicksort_1#_A(x1) = 0
              add_2_A(x1,x2) = x1 + x2 + 10
              high_2_A(x1,x2) = x1 + x2 + 9
              low_2_A(x1,x2) = x1 + x2 + 7
              le_2_A(x1,x2) = 3
              |0|_0_A() = 12
              true_0_A() = 3
              s_1_A(x1) = 6
              false_0_A() = 3
              nil_0_A() = 0
              if_low_3_A(x1,x2,x3) = x2 + x3 + 7
              if_high_3_A(x1,x2,x3) = x2 + x3 + 9
              quicksort_1#!_A(x1) = x1
              add_2!_A(x1,x2) = 8
              high_2!_A(x1,x2) = 7
              low_2!_A(x1,x2) = x1 + x2 + 13
              le_2!_A(x1,x2) = 11
              |0|_0!_A() = 13
              true_0!_A() = 12
              s_1!_A(x1) = 10
              false_0!_A() = 9
              nil_0!_A() = 0
              if_low_3!_A(x1,x2,x3) = x2 + x3 + 4
              if_high_3!_A(x1,x2,x3) = 6
        
  
    partial status:
  
      pi(quicksort_1#) = []
      pi(add_2) = [2]
      pi(high_2) = []
      pi(low_2) = [2]
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_low_3) = [3]
      pi(if_high_3) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  r44, r52

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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) = x2 + 5
              high_2_A(x1,x2) = x1 + 6
              nil_0_A() = 0
              filter_2_A(x1,x2) = x1 + 3
              app_2#!_A(x1,x2) = x1
              add_2!_A(x1,x2) = 8
              high_2!_A(x1,x2) = 2
              nil_0!_A() = 1
              filter_2!_A(x1,x2) = x1 + x2 + 12
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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 + 4
              high_2_A(x1,x2) = x1 + x2 + 5
              nil_0_A() = 4
              filter_2_A(x1,x2) = x1 + x2 + 3
              app_2#!_A(x1,x2) = x1 + 10
              add_2!_A(x1,x2) = x1 + 9
              high_2!_A(x1,x2) = x2 + 8
              nil_0!_A() = 6
              filter_2!_A(x1,x2) = 9
        
  
    partial status:
  
      pi(app_2#) = []
      pi(add_2) = []
      pi(high_2) = []
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: high_2#(n,add_2(m,x)) -> if_high_3#(le_2(m,n),n,add_2(m,x))
p2: if_high_3#(false_0(),n,add_2(m,x)) -> high_2#(n,x)
p3: if_high_3#(true_0(),n,add_2(m,x)) -> high_2#(n,x)

and R consists of:

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

The set of usable rules consists of

  r35, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              high_2#_A(x1,x2) = 0
              add_2_A(x1,x2) = 5
              if_high_3#_A(x1,x2,x3) = 0
              le_2_A(x1,x2) = x1 + x2 + 2
              false_0_A() = 4
              true_0_A() = 1
              |0|_0_A() = 4
              s_1_A(x1) = 4
              high_2#!_A(x1,x2) = 13
              add_2!_A(x1,x2) = 14
              if_high_3#!_A(x1,x2,x3) = 13
              le_2!_A(x1,x2) = x1 + x2 + 12
              false_0!_A() = 17
              true_0!_A() = 15
              |0|_0!_A() = 16
              s_1!_A(x1) = 18
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              high_2#_A(x1,x2) = 0
              add_2_A(x1,x2) = x1 + x2 + 4
              if_high_3#_A(x1,x2,x3) = 0
              le_2_A(x1,x2) = 2
              false_0_A() = 1
              true_0_A() = 1
              |0|_0_A() = 3
              s_1_A(x1) = x1 + 12
              high_2#!_A(x1,x2) = x2 + 3
              add_2!_A(x1,x2) = x2 + 5
              if_high_3#!_A(x1,x2,x3) = x3 + 2
              le_2!_A(x1,x2) = x1 + 8
              false_0!_A() = 11
              true_0!_A() = 9
              |0|_0!_A() = 10
              s_1!_A(x1) = 12
        
  
    partial status:
  
      pi(high_2#) = []
      pi(add_2) = [1]
      pi(if_high_3#) = []
      pi(le_2) = [1]
      pi(false_0) = []
      pi(true_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: if_low_3#(true_0(),n,add_2(m,x)) -> low_2#(n,x)
p2: low_2#(n,add_2(m,x)) -> if_low_3#(le_2(m,n),n,add_2(m,x))
p3: if_low_3#(false_0(),n,add_2(m,x)) -> low_2#(n,x)

and R consists of:

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

The set of usable rules consists of

  r35, r36, r37

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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() = 2
              add_2_A(x1,x2) = x1 + x2 + 8
              low_2#_A(x1,x2) = x2 + 8
              le_2_A(x1,x2) = x1 + x2 + 2
              false_0_A() = 4
              |0|_0_A() = 3
              s_1_A(x1) = x1 + 2
              if_low_3#!_A(x1,x2,x3) = x2 + x3
              true_0!_A() = 10
              add_2!_A(x1,x2) = 9
              low_2#!_A(x1,x2) = x1 + x2 + 7
              le_2!_A(x1,x2) = x2 + 14
              false_0!_A() = 12
              |0|_0!_A() = 11
              s_1!_A(x1) = 13
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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() = 8
              add_2_A(x1,x2) = x1 + x2 + 15
              low_2#_A(x1,x2) = x2 + 15
              le_2_A(x1,x2) = x1 + x2 + 5
              false_0_A() = 8
              |0|_0_A() = 6
              s_1_A(x1) = x1 + 6
              if_low_3#!_A(x1,x2,x3) = x1 + x3
              true_0!_A() = 0
              add_2!_A(x1,x2) = 21
              low_2#!_A(x1,x2) = 22
              le_2!_A(x1,x2) = x1 + x2 + 23
              false_0!_A() = 5
              |0|_0!_A() = 1
              s_1!_A(x1) = 24
        
  
    partial status:
  
      pi(if_low_3#) = [2, 3]
      pi(true_0) = []
      pi(add_2) = []
      pi(low_2#) = [1]
      pi(le_2) = []
      pi(false_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              le_2#_A(x1,x2) = x1 + x2
              s_1_A(x1) = x1 + 3
              |app'|_A(x1,x2) = x2 + 1
              if_low_1_A(x1) = 2
              if_low_2_A(x1,x2) = 1
              le_2#!_A(x1,x2) = x2
              s_1!_A(x1) = x1 + 1
              |app'|!_A(x1,x2) = 3
              if_low_1!_A(x1) = 1
              if_low_2!_A(x1,x2) = 2
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              le_2#_A(x1,x2) = 0
              s_1_A(x1) = x1 + 6
              |app'|_A(x1,x2) = x1 + 1
              if_low_1_A(x1) = x1 + 3
              if_low_2_A(x1,x2) = x1 + 1
              le_2#!_A(x1,x2) = x2 + 2
              s_1!_A(x1) = x1 + 3
              |app'|!_A(x1,x2) = 4
              if_low_1!_A(x1) = 0
              if_low_2!_A(x1,x2) = 5
        
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = []
      pi(|app'|) = [1]
      pi(if_low_1) = []
      pi(if_low_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

  r31, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              quot_2#_A(x1,x2) = x1
              s_1_A(x1) = x1 + 3
              minus_2_A(x1,x2) = x1 + 2
              |0|_0_A() = 4
              quot_2#!_A(x1,x2) = 3
              s_1!_A(x1) = 4
              minus_2!_A(x1,x2) = 5
              |0|_0!_A() = 12
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              quot_2#_A(x1,x2) = x1
              s_1_A(x1) = x1 + 7
              minus_2_A(x1,x2) = x1 + 5
              |0|_0_A() = 6
              quot_2#!_A(x1,x2) = x1 + x2 + 4
              s_1!_A(x1) = 17
              minus_2!_A(x1,x2) = 15
              |0|_0!_A() = 16
        
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = [1]
      pi(minus_2) = [1, 2]
      pi(|0|_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)

and R consists of:

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

The set of usable rules consists of

  r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              minus_2#_A(x1,x2) = x1 + x2
              s_1_A(x1) = x1 + 2
              |app'|_A(x1,x2) = x2 + 1
              if_low_1_A(x1) = 7
              if_low_2_A(x1,x2) = 1
              minus_2#!_A(x1,x2) = x1
              s_1!_A(x1) = 10
              |app'|!_A(x1,x2) = x1 + 1
              if_low_1!_A(x1) = 1
              if_low_2!_A(x1,x2) = 7
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              minus_2#_A(x1,x2) = 0
              s_1_A(x1) = 7
              |app'|_A(x1,x2) = x1 + 5
              if_low_1_A(x1) = x1 + 7
              if_low_2_A(x1,x2) = x1 + 1
              minus_2#!_A(x1,x2) = x2
              s_1!_A(x1) = 21
              |app'|!_A(x1,x2) = 3
              if_low_1!_A(x1) = 0
              if_low_2!_A(x1,x2) = 1
        
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = [1]
      pi(|app'|) = []
      pi(if_low_1) = []
      pi(if_low_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.