YES
We show the termination of the TRS R:

  |app'|(|app'|(eq(),|0|()),|0|()) -> true()
  |app'|(|app'|(eq(),|0|()),|app'|(s(),x)) -> false()
  |app'|(|app'|(eq(),|app'|(s(),x)),|0|()) -> false()
  |app'|(|app'|(eq(),|app'|(s(),x)),|app'|(s(),y)) -> |app'|(|app'|(eq(),x),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'|(min(),|app'|(|app'|(add(),n),nil())) -> n
  |app'|(min(),|app'|(|app'|(add(),n),|app'|(|app'|(add(),m),x))) -> |app'|(|app'|(if_min(),|app'|(|app'|(le(),n),m)),|app'|(|app'|(add(),n),|app'|(|app'|(add(),m),x)))
  |app'|(|app'|(if_min(),true()),|app'|(|app'|(add(),n),|app'|(|app'|(add(),m),x))) -> |app'|(min(),|app'|(|app'|(add(),n),x))
  |app'|(|app'|(if_min(),false()),|app'|(|app'|(add(),n),|app'|(|app'|(add(),m),x))) -> |app'|(min(),|app'|(|app'|(add(),m),x))
  |app'|(|app'|(rm(),n),nil()) -> nil()
  |app'|(|app'|(rm(),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(|app'|(if_rm(),|app'|(|app'|(eq(),n),m)),n),|app'|(|app'|(add(),m),x))
  |app'|(|app'|(|app'|(if_rm(),true()),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(rm(),n),x)
  |app'|(|app'|(|app'|(if_rm(),false()),n),|app'|(|app'|(add(),m),x)) -> |app'|(|app'|(add(),m),|app'|(|app'|(rm(),n),x))
  |app'|(|app'|(minsort(),nil()),nil()) -> nil()
  |app'|(|app'|(minsort(),|app'|(|app'|(add(),n),x)),y) -> |app'|(|app'|(|app'|(if_minsort(),|app'|(|app'|(eq(),n),|app'|(min(),|app'|(|app'|(add(),n),x)))),|app'|(|app'|(add(),n),x)),y)
  |app'|(|app'|(|app'|(if_minsort(),true()),|app'|(|app'|(add(),n),x)),y) -> |app'|(|app'|(add(),n),|app'|(|app'|(minsort(),|app'|(|app'|(app(),|app'|(|app'|(rm(),n),x)),y)),nil()))
  |app'|(|app'|(|app'|(if_minsort(),false()),|app'|(|app'|(add(),n),x)),y) -> |app'|(|app'|(minsort(),x),|app'|(|app'|(add(),n),y))
  |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'|(eq_0(),y) -> eq_1(y)
  |app'|(eq_1(x1),y) -> eq_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'|(if_min_0(),y) -> if_min_1(y)
  |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
  |app'|(if_minsort_0(),y) -> if_minsort_1(y)
  |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
  |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
  |app'|(if_rm_0(),y) -> if_rm_1(y)
  |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
  |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
  |app'|(le_0(),y) -> le_1(y)
  |app'|(le_1(x1),y) -> le_2(x1,y)
  |app'|(map_0(),y) -> map_1(y)
  |app'|(map_1(x1),y) -> map_2(x1,y)
  |app'|(min_0(),y) -> min_1(y)
  |app'|(minsort_0(),y) -> minsort_1(y)
  |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
  |app'|(rm_0(),y) -> rm_1(y)
  |app'|(rm_1(x1),y) -> rm_2(x1,y)
  |app'|(s_0(),y) -> s_1(y)
  eq_2(|0|_0(),|0|_0()) -> true_0()
  eq_2(|0|_0(),s_1(x)) -> false_0()
  eq_2(s_1(x),|0|_0()) -> false_0()
  eq_2(s_1(x),s_1(y)) -> eq_2(x,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))
  min_1(add_2(n,nil_0())) -> n
  min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
  if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
  if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
  rm_2(n,nil_0()) -> nil_0()
  rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
  if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
  if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
  minsort_2(nil_0(),nil_0()) -> nil_0()
  minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
  if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
  if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
  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'|#(eq_1(x1),y) -> eq_2#(x1,y)
p3: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p4: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: |app'|#(if_min_1(x1),y) -> if_min_2#(x1,y)
p6: |app'|#(if_minsort_2(x1,x2),y) -> if_minsort_3#(x1,x2,y)
p7: |app'|#(if_rm_2(x1,x2),y) -> if_rm_3#(x1,x2,y)
p8: |app'|#(le_1(x1),y) -> le_2#(x1,y)
p9: |app'|#(map_1(x1),y) -> map_2#(x1,y)
p10: |app'|#(min_0(),y) -> min_1#(y)
p11: |app'|#(minsort_1(x1),y) -> minsort_2#(x1,y)
p12: |app'|#(rm_1(x1),y) -> rm_2#(x1,y)
p13: eq_2#(s_1(x),s_1(y)) -> eq_2#(x,y)
p14: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)
p15: app_2#(add_2(n,x),y) -> app_2#(x,y)
p16: min_1#(add_2(n,add_2(m,x))) -> if_min_2#(le_2(n,m),add_2(n,add_2(m,x)))
p17: min_1#(add_2(n,add_2(m,x))) -> le_2#(n,m)
p18: if_min_2#(true_0(),add_2(n,add_2(m,x))) -> min_1#(add_2(n,x))
p19: if_min_2#(false_0(),add_2(n,add_2(m,x))) -> min_1#(add_2(m,x))
p20: rm_2#(n,add_2(m,x)) -> if_rm_3#(eq_2(n,m),n,add_2(m,x))
p21: rm_2#(n,add_2(m,x)) -> eq_2#(n,m)
p22: if_rm_3#(true_0(),n,add_2(m,x)) -> rm_2#(n,x)
p23: if_rm_3#(false_0(),n,add_2(m,x)) -> rm_2#(n,x)
p24: minsort_2#(add_2(n,x),y) -> if_minsort_3#(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
p25: minsort_2#(add_2(n,x),y) -> eq_2#(n,min_1(add_2(n,x)))
p26: minsort_2#(add_2(n,x),y) -> min_1#(add_2(n,x))
p27: if_minsort_3#(true_0(),add_2(n,x),y) -> minsort_2#(app_2(rm_2(n,x),y),nil_0())
p28: if_minsort_3#(true_0(),add_2(n,x),y) -> app_2#(rm_2(n,x),y)
p29: if_minsort_3#(true_0(),add_2(n,x),y) -> rm_2#(n,x)
p30: if_minsort_3#(false_0(),add_2(n,x),y) -> minsort_2#(x,add_2(n,y))
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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p3, p4, p9, p31, p32, p33, p34, p35, p36}
  {p24, p27, p30}
  {p15}
  {p20, p22, p23}
  {p13}
  {p16, p18, p19}
  {p14}


-- 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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r10, r11, r13, r14, r16, r18, r19, r21, r22, r23, r25, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r40, r41, r42, r43, r44, r48, r54

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 + 7
          add_2_A(x1,x2) = x1 + x2 + 6
          |app'|#_A(x1,x2) = x1 + x2 + 7
          map_1_A(x1) = x1 + 2
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 9
          false_0_A() = 3
          filter_2#_A(x1,x2) = x1 + x2 + 8
          filter_1_A(x1) = x1 + 2
          |app'|_A(x1,x2) = x1 + x2 + 7
          true_0_A() = 7
          add_0_A() = 2
          add_1_A(x1) = x1 + 1
          app_0_A() = 4
          app_1_A(x1) = x1 + 10
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 6
          eq_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 + 3
          if_min_0_A() = 1
          if_min_1_A(x1) = x1 + 2
          if_min_2_A(x1,x2) = x2 + 3
          if_minsort_1_A(x1) = x1 + 9
          if_minsort_2_A(x1,x2) = x1 + 1
          if_rm_0_A() = 1
          if_rm_1_A(x1) = x1 + 2
          if_rm_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 7
          map_0_A() = 1
          min_0_A() = 1
          min_1_A(x1) = x1 + 4
          minsort_0_A() = 1
          minsort_1_A(x1) = x1 + 3
          rm_0_A() = 1
          rm_1_A(x1) = x1 + 2
          s_0_A() = 3
          s_1_A(x1) = x1 + 4
          |0|_0_A() = 8
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 5
          minsort_2_A(x1,x2) = 2
          filter_2_A(x1,x2) = x1 + x2 + 8
  
    base order 2:
  
      precedence:
      
        |app'|# > filter2_4# = if_minsort_1 > filter_2# = if_min_2 = if_minsort_2 = filter_2 > map_2# = |app'| > map_1 = eq_2 = le_0 = le_2 = minsort_1 > add_2 = filter2_3 = filter_1 = app_1 = eq_1 = if_min_1 = le_1 = map_0 = minsort_0 = s_1 > add_1 = app_0 = eq_0 = filter_0 = filter2_2 = if_min_0 = min_1 = s_0 > add_0 = filter2_1 = min_0 = |0|_0 = rm_2 > false_0 = true_0 = filter2_0 = rm_1 > if_rm_2 = rm_0 = minsort_2 > if_rm_1 = nil_0 > if_rm_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(add_2) = [1, 2]
      pi(|app'|#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [2]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = []
      pi(|app'|) = []
      pi(true_0) = []
      pi(add_0) = []
      pi(add_1) = [1]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(eq_0) = []
      pi(eq_1) = [1]
      pi(eq_2) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(if_min_0) = []
      pi(if_min_1) = [1]
      pi(if_min_2) = [2]
      pi(if_minsort_1) = [1]
      pi(if_minsort_2) = [1]
      pi(if_rm_0) = []
      pi(if_rm_1) = [1]
      pi(if_rm_2) = [1, 2]
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(min_0) = []
      pi(min_1) = [1]
      pi(minsort_0) = []
      pi(minsort_1) = [1]
      pi(rm_0) = []
      pi(rm_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(nil_0) = []
      pi(rm_2) = []
      pi(minsort_2) = []
      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'|#(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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r9, r10, r11, r13, r14, r15, r16, r18, r19, r21, r22, r23, r25, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r48, r54

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 + 3
          add_2_A(x1,x2) = x1 + x2 + 6
          |app'|#_A(x1,x2) = x1 + x2 + 7
          filter_1_A(x1) = x1 + 2
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 7
          |app'|_A(x1,x2) = x1 + x2 + 7
          true_0_A() = 2
          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() = 2
          app_1_A(x1) = x1 + 1
          app_2_A(x1,x2) = x1 + x2 + 7
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 6
          eq_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 + 3
          if_min_0_A() = 2
          if_min_1_A(x1) = x1 + 1
          if_min_2_A(x1,x2) = x2 + 2
          if_minsort_0_A() = 1
          if_minsort_1_A(x1) = x1 + 2
          if_minsort_2_A(x1,x2) = x1 + x2 + 3
          if_rm_0_A() = 1
          if_rm_1_A(x1) = x1 + 2
          if_rm_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 7
          le_2_A(x1,x2) = x1 + x2 + 13
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          min_0_A() = 1
          min_1_A(x1) = x1 + 3
          minsort_0_A() = 1
          minsort_1_A(x1) = x1 + 2
          rm_0_A() = 1
          rm_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 3
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 5
          minsort_2_A(x1,x2) = x1 + x2 + 4
          filter_2_A(x1,x2) = x1 + x2 + 3
  
    base order 2:
  
      precedence:
      
        |app'|# > filter2_4# > filter_2# = filter_2 > |app'| > min_1 > app_2 = eq_2 = le_2 > add_2 = filter_1 = filter2_3 = app_1 = eq_1 = if_min_2 = le_1 = map_1 = min_0 = s_1 > add_1 = app_0 = eq_0 = filter_0 = filter2_2 = if_min_1 = le_0 = map_0 = s_0 > add_0 = filter2_1 = if_min_0 = if_minsort_2 = |0|_0 = rm_2 = minsort_2 > true_0 = false_0 = filter2_0 = if_minsort_1 = minsort_1 = rm_1 > if_minsort_0 = if_rm_2 = minsort_0 = rm_0 = nil_0 > if_rm_1 > if_rm_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(add_2) = [1, 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(eq_0) = []
      pi(eq_1) = [1]
      pi(eq_2) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(if_min_0) = []
      pi(if_min_1) = [1]
      pi(if_min_2) = [2]
      pi(if_minsort_0) = []
      pi(if_minsort_1) = [1]
      pi(if_minsort_2) = [2]
      pi(if_rm_0) = []
      pi(if_rm_1) = [1]
      pi(if_rm_2) = [1, 2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = [2]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(min_0) = []
      pi(min_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = [1]
      pi(rm_0) = []
      pi(rm_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(nil_0) = []
      pi(rm_2) = []
      pi(minsort_2) = []
      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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r9, r10, r11, r13, r14, r15, r18, r19, r21, r22, r23, r25, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r48, r54

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
          filter2_4#_A(x1,x2,x3,x4) = x4
          |app'|_A(x1,x2) = x1 + x2 + 2
          false_0_A() = 1
          true_0_A() = 7
          add_0_A() = 1
          add_1_A(x1) = x1 + 2
          app_0_A() = 4
          app_1_A(x1) = x1 + 5
          app_2_A(x1,x2) = x1 + x2 + 4
          eq_0_A() = 3
          eq_1_A(x1) = x1 + 4
          eq_2_A(x1,x2) = x1 + x2 + 5
          filter_0_A() = 6
          filter_1_A(x1) = x1 + 7
          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
          if_min_0_A() = 8
          if_min_1_A(x1) = x1 + 9
          if_min_2_A(x1,x2) = x2 + 8
          if_minsort_0_A() = 8
          if_minsort_1_A(x1) = 7
          if_rm_0_A() = 2
          if_rm_1_A(x1) = x1 + 1
          if_rm_2_A(x1,x2) = x1 + 2
          le_0_A() = 3
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 3
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          min_0_A() = 7
          min_1_A(x1) = x1 + 8
          minsort_0_A() = 7
          minsort_1_A(x1) = x1 + 8
          rm_0_A() = 2
          rm_1_A(x1) = x1 + 3
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 8
          nil_0_A() = 6
          rm_2_A(x1,x2) = x1 + x2 + 4
          minsort_2_A(x1,x2) = x1 + x2 + 9
          filter_2_A(x1,x2) = x1 + x2 + 8
  
    base order 2:
  
      precedence:
      
        filter_2 > |app'| > eq_1 > eq_2 > |0|_0 = minsort_2 > false_0 > app_2 = if_min_2 = le_2 = min_1 > true_0 = s_1 > app_0 = nil_0 > add_2 = app_1 = eq_0 = filter_1 = filter2_3 = if_min_1 = if_minsort_0 = le_1 = map_1 = min_0 = minsort_1 = s_0 > filter2_4# = add_1 = filter_0 = filter2_2 = if_min_0 = le_0 = map_0 = minsort_0 > filter_2# = add_0 = filter2_1 = rm_2 > filter2_0 = if_minsort_1 = rm_1 > if_rm_2 = rm_0 > if_rm_1 > if_rm_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(add_2) = [1, 2]
      pi(filter2_4#) = [4]
      pi(|app'|) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = [1]
      pi(eq_0) = []
      pi(eq_1) = [1]
      pi(eq_2) = [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(if_min_0) = []
      pi(if_min_1) = [1]
      pi(if_min_2) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = [1]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(min_0) = []
      pi(min_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = [1]
      pi(rm_0) = []
      pi(rm_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(nil_0) = []
      pi(rm_2) = []
      pi(minsort_2) = []
      pi(filter_2) = [2]

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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r9, r10, r11, r13, r14, r15, r16, r18, r19, r21, r22, r23, r25, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r48, r54

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 + 1
          |app'|_A(x1,x2) = x1 + x2 + 2
          false_0_A() = 1
          add_0_A() = 6
          add_1_A(x1) = x1 + 7
          app_0_A() = 9
          app_1_A(x1) = x1 + 10
          app_2_A(x1,x2) = x1 + x2 + 9
          eq_0_A() = 14
          eq_1_A(x1) = x1 + 15
          eq_2_A(x1,x2) = x1 + x2 + 16
          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
          if_min_0_A() = 1
          if_min_1_A(x1) = x1 + 2
          if_min_2_A(x1,x2) = x2 + 3
          if_minsort_0_A() = 1
          if_minsort_1_A(x1) = x1 + 2
          if_minsort_2_A(x1,x2) = x1 + x2 + 3
          if_rm_0_A() = 1
          if_rm_1_A(x1) = x1 + 2
          if_rm_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 7
          le_1_A(x1) = x1 + 8
          le_2_A(x1,x2) = x1 + x2 + 9
          map_0_A() = 7
          map_1_A(x1) = x1 + 8
          min_0_A() = 3
          min_1_A(x1) = x1 + 4
          minsort_0_A() = 3
          minsort_1_A(x1) = x1 + 4
          rm_0_A() = 5
          rm_1_A(x1) = x1 + 6
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 3
          true_0_A() = 2
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 7
          minsort_2_A(x1,x2) = x1 + x2 + 5
          filter_2_A(x1,x2) = x1 + x2 + 3
  
    base order 2:
  
      precedence:
      
        eq_2 > app_2 = |0|_0 = minsort_2 = filter_2 > add_2 = false_0 = if_min_2 = le_2 = min_1 > |app'| > filter_2# = add_1 = app_1 = eq_1 = filter_1 = filter2_3 = if_min_1 = if_minsort_2 = le_1 = map_1 = min_0 = minsort_1 = s_1 = rm_2 > filter2_4# = add_0 = app_0 = eq_0 = filter_0 = filter2_2 = if_min_0 = if_minsort_1 = le_0 = map_0 = minsort_0 = rm_1 = s_0 = true_0 > filter2_1 = if_minsort_0 = if_rm_2 = rm_0 = nil_0 > filter2_0 = if_rm_1 > if_rm_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(add_2) = []
      pi(filter2_4#) = []
      pi(|app'|) = [1]
      pi(false_0) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = []
      pi(eq_0) = []
      pi(eq_1) = [1]
      pi(eq_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [3]
      pi(if_min_0) = []
      pi(if_min_1) = [1]
      pi(if_min_2) = [2]
      pi(if_minsort_0) = []
      pi(if_minsort_1) = [1]
      pi(if_minsort_2) = [2]
      pi(if_rm_0) = []
      pi(if_rm_1) = [1]
      pi(if_rm_2) = [1, 2]
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(min_0) = []
      pi(min_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = [1]
      pi(rm_0) = []
      pi(rm_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(nil_0) = []
      pi(rm_2) = []
      pi(minsort_2) = [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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r11, r13, r14, r15, r16, r18, r19, r21, r23, r25, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r48, r54

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) = x2 + 15
          add_0_A() = 1
          add_1_A(x1) = 12
          app_0_A() = 2
          app_1_A(x1) = 1
          eq_0_A() = 2
          eq_1_A(x1) = 1
          filter_0_A() = 15
          filter_1_A(x1) = 14
          filter2_0_A() = 4
          filter2_1_A(x1) = x1 + 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          if_min_0_A() = 2
          if_min_1_A(x1) = 1
          if_min_2_A(x1,x2) = x2 + 2
          if_minsort_0_A() = 3
          if_minsort_1_A(x1) = 2
          if_minsort_2_A(x1,x2) = 1
          if_rm_0_A() = 3
          if_rm_1_A(x1) = 2
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          min_0_A() = 1
          min_1_A(x1) = x1 + 3
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          eq_2_A(x1,x2) = x1 + x2 + 16
          |0|_0_A() = 3
          true_0_A() = 2
          false_0_A() = 1
          le_2_A(x1,x2) = x1 + x2 + 16
          app_2_A(x1,x2) = x1 + x2 + 2
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 6
          minsort_2_A(x1,x2) = x1 + x2 + 17
          filter_2_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        filter_0 > filter_1 = if_rm_2 = |0|_0 = app_2 = filter_2 > map_2# = |app'| = app_0 = if_minsort_0 = map_0 = false_0 > add_2 = app_1 = eq_0 = filter2_1 = if_min_0 = if_minsort_1 = if_rm_1 = le_0 = map_1 = minsort_0 = rm_0 > add_1 = filter2_0 = filter2_2 = if_rm_0 > add_0 = filter2_3 = if_minsort_2 = rm_2 > rm_1 = eq_2 > eq_1 = minsort_2 > if_min_2 = min_1 = minsort_1 > if_min_1 = min_0 = le_2 > le_1 = s_1 = true_0 > s_0 = nil_0
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(add_2) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_min_0) = []
      pi(if_min_1) = []
      pi(if_min_2) = [2]
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(min_0) = []
      pi(min_1) = [1]
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(eq_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(le_2) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(rm_2) = []
      pi(minsort_2) = []
      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: minsort_2#(add_2(n,x),y) -> if_minsort_3#(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
p2: if_minsort_3#(false_0(),add_2(n,x),y) -> minsort_2#(x,add_2(n,y))
p3: if_minsort_3#(true_0(),add_2(n,x),y) -> minsort_2#(app_2(rm_2(n,x),y),nil_0())

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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r11, r14, r15, r16, r18, r19, r21, r23, r25, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r54

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:
          minsort_2#_A(x1,x2) = x1 + x2 + 5
          add_2_A(x1,x2) = x1 + x2 + 16
          if_minsort_3#_A(x1,x2,x3) = x2 + x3 + 5
          eq_2_A(x1,x2) = x1 + x2 + 17
          min_1_A(x1) = x1 + 4
          false_0_A() = 1
          true_0_A() = 15
          app_2_A(x1,x2) = x1 + x2 + 2
          rm_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
          |app'|_A(x1,x2) = x2 + 5
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 2
          app_1_A(x1) = 1
          eq_0_A() = 2
          eq_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
          if_min_1_A(x1) = 6
          if_min_2_A(x1,x2) = x2 + 3
          if_minsort_0_A() = 3
          if_minsort_1_A(x1) = 2
          if_minsort_2_A(x1,x2) = 1
          if_rm_0_A() = 3
          if_rm_1_A(x1) = 2
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          min_0_A() = 1
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 16
          le_2_A(x1,x2) = x1 + x2 + 17
          if_rm_3_A(x1,x2,x3) = x2 + x3 + 6
          minsort_2_A(x1,x2) = x1 + x2 + 8
          filter_2_A(x1,x2) = x1 + x2 + 16
  
    base order 2:
  
      precedence:
      
        filter2_0 > filter2_1 > rm_2 = filter2_2 = s_1 = if_rm_3 > eq_2 = app_2 = filter2_3 = if_min_2 > minsort_2# = add_2 = if_minsort_3# = if_min_1 > |app'| = add_0 = app_1 = eq_0 = filter_0 = if_minsort_0 = if_rm_0 = le_0 = map_0 = minsort_0 = rm_0 = s_0 = filter_2 > add_1 = app_0 = eq_1 = filter_1 = if_minsort_1 = if_rm_1 = map_1 = rm_1 = minsort_2 > min_1 = if_minsort_2 = if_rm_2 = minsort_1 > min_0 = |0|_0 = le_2 > false_0 = true_0 = le_1 > nil_0
      
  
    partial status:
  
      pi(minsort_2#) = []
      pi(add_2) = []
      pi(if_minsort_3#) = []
      pi(eq_2) = []
      pi(min_1) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(app_2) = []
      pi(rm_2) = [2]
      pi(nil_0) = []
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_min_1) = []
      pi(if_min_2) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(min_0) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(le_2) = []
      pi(if_rm_3) = [3]
      pi(minsort_2) = [2]
      pi(filter_2) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: minsort_2#(add_2(n,x),y) -> if_minsort_3#(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
p2: if_minsort_3#(false_0(),add_2(n,x),y) -> minsort_2#(x,add_2(n,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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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: minsort_2#(add_2(n,x),y) -> if_minsort_3#(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
p2: if_minsort_3#(false_0(),add_2(n,x),y) -> minsort_2#(x,add_2(n,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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r11, r13, r15, r16, r18, r19, r21, r23, r26, r28, r31, r32, r33, r34, r35, r36, r37, r38, r44

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:
          minsort_2#_A(x1,x2) = x1
          add_2_A(x1,x2) = x2
          if_minsort_3#_A(x1,x2,x3) = x1 + x2
          eq_2_A(x1,x2) = 0
          min_1_A(x1) = 0
          false_0_A() = 0
          |app'|_A(x1,x2) = 5
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 5
          app_1_A(x1) = 4
          eq_0_A() = 2
          eq_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
          if_min_0_A() = 3
          if_min_1_A(x1) = 2
          if_minsort_0_A() = 1
          if_minsort_1_A(x1) = 4
          if_minsort_2_A(x1,x2) = 3
          if_rm_0_A() = 3
          if_rm_1_A(x1) = 2
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          |0|_0_A() = 1
          true_0_A() = 0
          s_1_A(x1) = x1 + 6
          le_2_A(x1,x2) = x1 + 7
          app_2_A(x1,x2) = x2 + 3
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      precedence:
      
        filter2_0 > filter2_1 > app_0 = filter_0 = filter2_2 = if_min_0 = map_0 = app_2 > min_1 = add_1 = app_1 = filter_1 = filter2_3 = if_min_1 = map_1 = nil_0 > minsort_2# = add_2 = |app'| = add_0 = eq_0 = if_minsort_1 = if_minsort_2 = if_rm_0 = minsort_0 = rm_0 > if_minsort_3# = eq_2 = eq_1 = if_minsort_0 = if_rm_1 = |0|_0 > true_0 > le_0 = le_2 > false_0 = le_1 = minsort_1 > s_1 > rm_2 > rm_1 > if_rm_2
      
  
    partial status:
  
      pi(minsort_2#) = [1]
      pi(add_2) = [2]
      pi(if_minsort_3#) = [2]
      pi(eq_2) = []
      pi(min_1) = []
      pi(false_0) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_min_0) = []
      pi(if_min_1) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(le_2) = []
      pi(app_2) = [2]
      pi(nil_0) = []
      pi(rm_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: if_minsort_3#(false_0(),add_2(n,x),y) -> minsort_2#(x,add_2(n,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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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: 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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r5, r7, r9, r11, r13, r15, r16, r18, r19, r21, r22, r23, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r44, r48, r54

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 + x2 + 5
          add_2_A(x1,x2) = x1 + x2 + 4
          |app'|_A(x1,x2) = x2 + 6
          add_0_A() = 6
          add_1_A(x1) = 5
          eq_0_A() = 2
          eq_1_A(x1) = 1
          filter_0_A() = 5
          filter_1_A(x1) = 4
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 5
          filter2_2_A(x1,x2) = x1 + 4
          filter2_3_A(x1,x2,x3) = 2
          if_min_0_A() = 2
          if_min_1_A(x1) = 1
          if_minsort_0_A() = 3
          if_minsort_1_A(x1) = 2
          if_minsort_2_A(x1,x2) = 1
          if_rm_0_A() = 2
          if_rm_1_A(x1) = 1
          if_rm_2_A(x1,x2) = x2 + 2
          le_0_A() = 2
          le_1_A(x1) = 1
          le_2_A(x1,x2) = x2 + 3
          map_0_A() = 2
          map_1_A(x1) = 1
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 4
          eq_2_A(x1,x2) = x1 + 3
          |0|_0_A() = 3
          true_0_A() = 2
          false_0_A() = 2
          app_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 3
          minsort_2_A(x1,x2) = x1 + x2 + 8
          filter_2_A(x1,x2) = x1 + x2 + 3
  
    base order 2:
  
      precedence:
      
        filter2_2 = |0|_0 = app_2 = filter_2 > filter2_3 = if_minsort_0 = true_0 > app_2# = if_minsort_1 = if_rm_0 = minsort_2 > add_2 = if_min_0 = if_minsort_2 = if_rm_1 = eq_2 = rm_2 > |app'| = add_0 = eq_0 = filter_0 = if_min_1 = le_0 = map_0 = minsort_0 = rm_0 = nil_0 > add_1 = eq_1 = filter_1 = filter2_1 = le_2 = map_1 = minsort_1 = rm_1 = s_1 > filter2_0 = if_rm_2 = le_1 = s_0 = false_0
      
  
    partial status:
  
      pi(app_2#) = [2]
      pi(add_2) = [1, 2]
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1]
      pi(filter2_3) = []
      pi(if_min_0) = []
      pi(if_min_1) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = [2]
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(eq_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(app_2) = [2]
      pi(nil_0) = []
      pi(rm_2) = []
      pi(minsort_2) = []
      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: if_rm_3#(false_0(),n,add_2(m,x)) -> rm_2#(n,x)
p2: rm_2#(n,add_2(m,x)) -> if_rm_3#(eq_2(n,m),n,add_2(m,x))
p3: if_rm_3#(true_0(),n,add_2(m,x)) -> rm_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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r11, r13, r14, r15, r18, r19, r21, r23, r25, r26, r28, r30, r31, r32, r33, r34, r35, r36, r37, r40, r41, r42, r43, r44, r48, r54

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_rm_3#_A(x1,x2,x3) = x3
          false_0_A() = 2
          add_2_A(x1,x2) = x1 + x2 + 7
          rm_2#_A(x1,x2) = x2 + 1
          eq_2_A(x1,x2) = x1 + x2 + 12
          true_0_A() = 9
          |app'|_A(x1,x2) = x2 + 16
          add_0_A() = 2
          add_1_A(x1) = 1
          app_0_A() = 2
          app_1_A(x1) = 1
          eq_0_A() = 2
          eq_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
          if_min_0_A() = 2
          if_min_1_A(x1) = 1
          if_min_2_A(x1,x2) = x2 + 2
          if_minsort_0_A() = 2
          if_minsort_1_A(x1) = 1
          if_rm_0_A() = 1
          if_rm_1_A(x1) = x1 + 2
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          min_0_A() = 1
          min_1_A(x1) = x1 + 6
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 11
          |0|_0_A() = 10
          le_2_A(x1,x2) = x1 + 10
          nil_0_A() = 8
          rm_2_A(x1,x2) = x1 + x2 + 6
          minsort_2_A(x1,x2) = x1 + x2 + 17
          filter_2_A(x1,x2) = x1 + x2 + 10
  
    base order 2:
  
      precedence:
      
        rm_2 > eq_2 = |0|_0 = filter_2 > true_0 > eq_0 = nil_0 = minsort_2 > add_0 = eq_1 = filter_0 = if_rm_1 > add_2 = |app'| = add_1 = app_0 = filter_1 = filter2_0 = if_min_0 = if_minsort_0 = if_rm_0 = le_0 = map_0 = minsort_0 = rm_0 = le_2 > if_rm_3# = app_1 = filter2_1 = if_min_2 = if_minsort_1 = if_rm_2 = le_1 = map_1 = min_1 = minsort_1 = rm_1 > rm_2# = filter2_2 = if_min_1 = min_0 > false_0 = filter2_3 > s_1 > s_0
      
  
    partial status:
  
      pi(if_rm_3#) = []
      pi(false_0) = []
      pi(add_2) = [2]
      pi(rm_2#) = [2]
      pi(eq_2) = []
      pi(true_0) = []
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_min_0) = []
      pi(if_min_1) = []
      pi(if_min_2) = [2]
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = [1]
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(min_0) = []
      pi(min_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(le_2) = []
      pi(nil_0) = []
      pi(rm_2) = [2]
      pi(minsort_2) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: rm_2#(n,add_2(m,x)) -> if_rm_3#(eq_2(n,m),n,add_2(m,x))
p2: if_rm_3#(true_0(),n,add_2(m,x)) -> rm_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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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: rm_2#(n,add_2(m,x)) -> if_rm_3#(eq_2(n,m),n,add_2(m,x))
p2: if_rm_3#(true_0(),n,add_2(m,x)) -> rm_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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r11, r13, r15, r16, r18, r19, r21, r23, r26, r28, r30, r32, r33, r35, r36, r37, r40, r44, r48, r54

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:
          rm_2#_A(x1,x2) = x2 + 1
          add_2_A(x1,x2) = x1 + x2 + 4
          if_rm_3#_A(x1,x2,x3) = x3
          eq_2_A(x1,x2) = x1 + x2 + 2
          true_0_A() = 7
          |app'|_A(x1,x2) = x2 + 11
          add_0_A() = 6
          add_1_A(x1) = 5
          app_0_A() = 2
          app_1_A(x1) = 1
          eq_0_A() = 4
          eq_1_A(x1) = 3
          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
          if_min_0_A() = 2
          if_min_1_A(x1) = 1
          if_minsort_0_A() = 11
          if_minsort_1_A(x1) = 10
          if_minsort_2_A(x1,x2) = 9
          if_rm_0_A() = 12
          if_rm_1_A(x1) = 11
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 5
          rm_1_A(x1) = 4
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 2
          false_0_A() = 1
          le_2_A(x1,x2) = x1 + x2 + 7
          min_1_A(x1) = x1 + 5
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 3
          minsort_2_A(x1,x2) = x1 + x2 + 14
          filter_2_A(x1,x2) = x1 + x2 + 2
  
    base order 2:
  
      precedence:
      
        le_2 = minsort_2 > add_2 = add_0 = app_0 = if_rm_0 = rm_0 = |0|_0 = min_1 > |app'| = add_1 = app_1 = eq_0 = filter_0 = filter2_0 = if_min_0 = if_minsort_0 = le_0 = map_0 = minsort_0 = filter_2 > eq_1 = filter_1 = filter2_1 = if_min_1 = if_minsort_1 = if_rm_1 = le_1 = map_1 = minsort_1 > rm_2# = eq_2 = filter2_2 = if_minsort_2 > if_rm_3# = filter2_3 > s_1 = rm_2 > true_0 = rm_1 = s_0 = false_0 > if_rm_2 = nil_0
      
  
    partial status:
  
      pi(rm_2#) = [2]
      pi(add_2) = [2]
      pi(if_rm_3#) = []
      pi(eq_2) = [2]
      pi(true_0) = []
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_min_0) = []
      pi(if_min_1) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(false_0) = []
      pi(le_2) = []
      pi(min_1) = [1]
      pi(nil_0) = []
      pi(rm_2) = [1, 2]
      pi(minsort_2) = []
      pi(filter_2) = [2]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: rm_2#(n,add_2(m,x)) -> if_rm_3#(eq_2(n,m),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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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: eq_2#(s_1(x),s_1(y)) -> eq_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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r5, r10, r11, r13, r15, r16, r18, r19, r21, r23, r26, r28, r31, r32, r33, r34, r35, r36, r37, r38, r39, r44, r48, r54

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:
          eq_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 6
          |app'|_A(x1,x2) = 4
          add_0_A() = 2
          add_1_A(x1) = 1
          eq_0_A() = 2
          eq_1_A(x1) = 1
          filter2_1_A(x1) = x1 + 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          if_min_0_A() = 2
          if_min_1_A(x1) = 1
          if_minsort_0_A() = 3
          if_minsort_1_A(x1) = 2
          if_minsort_2_A(x1,x2) = 1
          if_rm_0_A() = 3
          if_rm_1_A(x1) = 2
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          eq_2_A(x1,x2) = x1 + 5
          |0|_0_A() = 3
          true_0_A() = 2
          false_0_A() = 2
          le_2_A(x1,x2) = x1 + x2 + 7
          app_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
          add_2_A(x1,x2) = x1 + x2 + 8
          rm_2_A(x1,x2) = x1 + x2 + 5
          minsort_2_A(x1,x2) = x1 + x2 + 6
          filter_2_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        app_2 > eq_2# = filter2_1 = if_minsort_0 = if_rm_0 = le_0 = eq_2 = |0|_0 = add_2 = rm_2 = minsort_2 > filter_2 > s_1 = |app'| = add_0 = eq_0 = if_min_0 = map_0 = minsort_0 = rm_0 = le_2 > add_1 = eq_1 = filter2_2 = if_min_1 = if_minsort_1 = if_rm_2 = le_1 = map_1 = minsort_1 = rm_1 = true_0 = false_0 > filter2_3 = if_minsort_2 = if_rm_1 = nil_0
      
  
    partial status:
  
      pi(eq_2#) = []
      pi(s_1) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_min_0) = []
      pi(if_min_1) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(eq_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(le_2) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(add_2) = [2]
      pi(rm_2) = []
      pi(minsort_2) = []
      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: if_min_2#(true_0(),add_2(n,add_2(m,x))) -> min_1#(add_2(n,x))
p2: min_1#(add_2(n,add_2(m,x))) -> if_min_2#(le_2(n,m),add_2(n,add_2(m,x)))
p3: if_min_2#(false_0(),add_2(n,add_2(m,x))) -> min_1#(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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r11, r15, r16, r18, r19, r21, r23, r26, r28, r30, r31, r32, r33, r44, r48, r54

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_min_2#_A(x1,x2) = x2 + 1
          true_0_A() = 16
          add_2_A(x1,x2) = x1 + x2 + 26
          min_1#_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 2
          false_0_A() = 29
          |app'|_A(x1,x2) = x2 + 19
          add_0_A() = 1
          add_1_A(x1) = 2
          app_0_A() = 3
          app_1_A(x1) = 2
          eq_0_A() = 19
          eq_1_A(x1) = 18
          filter_0_A() = 19
          filter_1_A(x1) = 18
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          if_minsort_0_A() = 2
          if_minsort_1_A(x1) = 1
          if_minsort_2_A(x1,x2) = x2 + 18
          if_rm_0_A() = 3
          if_rm_1_A(x1) = 2
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = 13
          eq_2_A(x1,x2) = x1 + 17
          |0|_0_A() = 13
          rm_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 0
          minsort_2_A(x1,x2) = x1 + x2 + 20
          filter_2_A(x1,x2) = 17
  
    base order 2:
  
      precedence:
      
        eq_2 = |0|_0 > if_min_2# = false_0 > min_1# = add_1 = app_0 = filter2_3 > true_0 = add_2 = le_2 = add_0 = app_1 = filter_0 = filter2_0 = if_minsort_0 = if_rm_0 = le_0 = rm_2 = minsort_2 > rm_0 > |app'| = eq_0 = le_1 = map_0 = minsort_0 = rm_1 > eq_1 = filter_1 = filter2_1 = if_minsort_2 = if_rm_1 = map_1 = minsort_1 = s_1 > filter2_2 = if_minsort_1 = if_rm_2 = s_0 = filter_2 > nil_0
      
  
    partial status:
  
      pi(if_min_2#) = [2]
      pi(true_0) = []
      pi(add_2) = []
      pi(min_1#) = [1]
      pi(le_2) = []
      pi(false_0) = []
      pi(|app'|) = [2]
      pi(add_0) = []
      pi(add_1) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = [2]
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(eq_2) = []
      pi(|0|_0) = []
      pi(rm_2) = [2]
      pi(nil_0) = []
      pi(minsort_2) = []
      pi(filter_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: if_min_2#(true_0(),add_2(n,add_2(m,x))) -> min_1#(add_2(n,x))
p2: if_min_2#(false_0(),add_2(n,add_2(m,x))) -> min_1#(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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: 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'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,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: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r5, r10, r11, r13, r15, r16, r18, r19, r21, r23, r26, r28, r31, r32, r33, r34, r35, r36, r37, r38, r39, r44, r48, r54

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 + 7
          s_1_A(x1) = x1 + 6
          |app'|_A(x1,x2) = 4
          add_0_A() = 2
          add_1_A(x1) = 1
          eq_0_A() = 2
          eq_1_A(x1) = 1
          filter2_1_A(x1) = x1 + 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          if_min_0_A() = 2
          if_min_1_A(x1) = 1
          if_minsort_0_A() = 3
          if_minsort_1_A(x1) = 2
          if_minsort_2_A(x1,x2) = 1
          if_rm_0_A() = 3
          if_rm_1_A(x1) = 2
          if_rm_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minsort_0_A() = 2
          minsort_1_A(x1) = 1
          rm_0_A() = 2
          rm_1_A(x1) = 1
          eq_2_A(x1,x2) = x1 + 5
          |0|_0_A() = 3
          true_0_A() = 2
          false_0_A() = 2
          le_2_A(x1,x2) = x1 + x2 + 7
          app_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
          add_2_A(x1,x2) = x1 + x2 + 8
          rm_2_A(x1,x2) = x1 + x2 + 5
          minsort_2_A(x1,x2) = x1 + x2 + 6
          filter_2_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        app_2 > le_2# = filter2_1 = if_minsort_0 = if_rm_0 = le_0 = eq_2 = |0|_0 = add_2 = rm_2 = minsort_2 > filter_2 > s_1 = |app'| = add_0 = eq_0 = if_min_0 = map_0 = minsort_0 = rm_0 = le_2 > add_1 = eq_1 = filter2_2 = if_min_1 = if_minsort_1 = if_rm_2 = le_1 = map_1 = minsort_1 = rm_1 = true_0 = false_0 > filter2_3 = if_minsort_2 = if_rm_1 = nil_0
      
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = []
      pi(|app'|) = []
      pi(add_0) = []
      pi(add_1) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_min_0) = []
      pi(if_min_1) = []
      pi(if_minsort_0) = []
      pi(if_minsort_1) = []
      pi(if_minsort_2) = []
      pi(if_rm_0) = []
      pi(if_rm_1) = []
      pi(if_rm_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minsort_0) = []
      pi(minsort_1) = []
      pi(rm_0) = []
      pi(rm_1) = []
      pi(eq_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(le_2) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(add_2) = [2]
      pi(rm_2) = []
      pi(minsort_2) = []
      pi(filter_2) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.