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

  r38, r39, r44, r48, r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x1 + x2 + 8
          add_2_A(x1,x2) = x1 + x2 + 7
          |app'|#_A(x1,x2) = x1 + x2 + 8
          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 + 6
          false_0_A() = 6
          filter_2#_A(x1,x2) = x1 + x2 + 5
          filter_1_A(x1) = x1 + 2
          |app'|_A(x1,x2) = x1 + x2 + 11
          true_0_A() = 2
          app_2_A(x1,x2) = x1 + x2 + 8
          nil_0_A() = 1
          rm_2_A(x1,x2) = x2 + 9
          minsort_2_A(x1,x2) = x1 + x2 + 10
          filter_2_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            map_2#_A(x1,x2) = x1 + x2
            add_2_A(x1,x2) = x1 + x2 + 21
            |app'|#_A(x1,x2) = x1 + x2
            map_1_A(x1) = x1 + 3
            filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
            filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 7
            false_0_A() = 10
            filter_2#_A(x1,x2) = x1 + x2 + 7
            filter_1_A(x1) = x1 + 7
            |app'|_A(x1,x2) = x1 + x2 + 1
            true_0_A() = 8
            app_2_A(x1,x2) = x1 + x2 + 5
            nil_0_A() = 6
            rm_2_A(x1,x2) = x1 + x2 + 8
            minsort_2_A(x1,x2) = x1 + x2 + 5
            filter_2_A(x1,x2) = x1 + x2 + 10
            map_2#!_A(x1,x2) = x2 + 21
            add_2!_A(x1,x2) = x1 + 18
            |app'|#!_A(x1,x2) = x1 + 17
            map_1!_A(x1) = 1
            filter2_3!_A(x1,x2,x3) = x3 + 17
            filter2_4#!_A(x1,x2,x3,x4) = 18
            false_0!_A() = 17
            filter_2#!_A(x1,x2) = x2 + 23
            filter_1!_A(x1) = 24
            |app'|!_A(x1,x2) = 44
            true_0!_A() = 22
            app_2!_A(x1,x2) = x1 + 59
            nil_0!_A() = 21
            rm_2!_A(x1,x2) = x1 + 20
            minsort_2!_A(x1,x2) = x1 + x2 + 46
            filter_2!_A(x1,x2) = x2 + 25
  
    partial status:
  
      pi(map_2#) = [1, 2]
      pi(add_2) = [2]
      pi(|app'|#) = [2]
      pi(map_1) = []
      pi(filter2_3) = []
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = []
      pi(|app'|) = [2]
      pi(true_0) = []
      pi(app_2) = [1]
      pi(nil_0) = []
      pi(rm_2) = []
      pi(minsort_2) = []
      pi(filter_2) = []

The following rules are removed:

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

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r38, r39, r44, r45, r46, r47, r48, r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minsort_2#_A(x1,x2) = x1 + x2 + 31
          add_2_A(x1,x2) = x2 + 13
          if_minsort_3#_A(x1,x2,x3) = x2 + x3 + 31
          eq_2_A(x1,x2) = x1 + x2 + 36
          min_1_A(x1) = 7
          false_0_A() = 1
          true_0_A() = 6
          app_2_A(x1,x2) = x1 + x2 + 7
          rm_2_A(x1,x2) = x2
          nil_0_A() = 1
          if_rm_3_A(x1,x2,x3) = x3
          minsort_2_A(x1,x2) = x1 + x2 + 29
          filter_2_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            minsort_2#_A(x1,x2) = x1 + x2
            add_2_A(x1,x2) = x1 + x2 + 18
            if_minsort_3#_A(x1,x2,x3) = x2 + x3
            eq_2_A(x1,x2) = x1 + x2 + 5
            min_1_A(x1) = x1 + 3
            false_0_A() = 10
            true_0_A() = 8
            app_2_A(x1,x2) = x1 + x2 + 5
            rm_2_A(x1,x2) = x1 + x2 + 7
            nil_0_A() = 6
            if_rm_3_A(x1,x2,x3) = x2 + x3 + 7
            minsort_2_A(x1,x2) = x1 + x2 + 5
            filter_2_A(x1,x2) = x1 + x2 + 10
            minsort_2#!_A(x1,x2) = x1 + 3
            add_2!_A(x1,x2) = 31
            if_minsort_3#!_A(x1,x2,x3) = x2
            eq_2!_A(x1,x2) = x1 + x2 + 68
            min_1!_A(x1) = 70
            false_0!_A() = 71
            true_0!_A() = 16
            app_2!_A(x1,x2) = x1 + x2 + 14
            rm_2!_A(x1,x2) = x1 + 33
            nil_0!_A() = 17
            if_rm_3!_A(x1,x2,x3) = x2 + 33
            minsort_2!_A(x1,x2) = x1 + x2 + 33
            filter_2!_A(x1,x2) = x2 + 12
  
    partial status:
  
      pi(minsort_2#) = [1]
      pi(add_2) = [2]
      pi(if_minsort_3#) = [2]
      pi(eq_2) = [1, 2]
      pi(min_1) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(app_2) = [1]
      pi(rm_2) = [2]
      pi(nil_0) = []
      pi(if_rm_3) = []
      pi(minsort_2) = [1]
      pi(filter_2) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r44, r48, r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app_2#_A(x1,x2) = 1
          add_2_A(x1,x2) = x1 + 4
          rm_2_A(x1,x2) = 2
          nil_0_A() = 1
          minsort_2_A(x1,x2) = x1 + x2 + 10
          filter_2_A(x1,x2) = x2 + 6
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            app_2#_A(x1,x2) = x1
            add_2_A(x1,x2) = x1 + x2 + 21
            rm_2_A(x1,x2) = x2
            nil_0_A() = 6
            minsort_2_A(x1,x2) = x2 + 5
            filter_2_A(x1,x2) = x1 + x2 + 10
            app_2#!_A(x1,x2) = x1
            add_2!_A(x1,x2) = x1 + x2 + 9
            rm_2!_A(x1,x2) = x1 + 7
            nil_0!_A() = 10
            minsort_2!_A(x1,x2) = x1 + x2 + 10
            filter_2!_A(x1,x2) = x2 + 12
  
    partial status:
  
      pi(app_2#) = []
      pi(add_2) = [1]
      pi(rm_2) = []
      pi(nil_0) = []
      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: 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

  r31, r32, r33, r34, r44, r48, r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_rm_3#_A(x1,x2,x3) = x2 + x3 + 2
          false_0_A() = 1
          add_2_A(x1,x2) = x1 + x2 + 8
          rm_2#_A(x1,x2) = x1 + x2 + 9
          eq_2_A(x1,x2) = x1 + x2 + 11
          true_0_A() = 3
          |0|_0_A() = 4
          s_1_A(x1) = x1 + 2
          rm_2_A(x1,x2) = x2 + 2
          nil_0_A() = 1
          minsort_2_A(x1,x2) = x1 + 7
          filter_2_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        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() = 10
            add_2_A(x1,x2) = x1 + x2 + 5
            rm_2#_A(x1,x2) = x2
            eq_2_A(x1,x2) = x1 + x2 + 6
            true_0_A() = 7
            |0|_0_A() = 1
            s_1_A(x1) = x1 + 3
            rm_2_A(x1,x2) = x1 + x2 + 8
            nil_0_A() = 7
            minsort_2_A(x1,x2) = x1 + x2 + 5
            filter_2_A(x1,x2) = x1 + x2 + 15
            if_rm_3#!_A(x1,x2,x3) = 0
            false_0!_A() = 11
            add_2!_A(x1,x2) = x1 + x2 + 14
            rm_2#!_A(x1,x2) = 1
            eq_2!_A(x1,x2) = x1 + x2 + 9
            true_0!_A() = 9
            |0|_0!_A() = 12
            s_1!_A(x1) = 22
            rm_2!_A(x1,x2) = 10
            nil_0!_A() = 0
            minsort_2!_A(x1,x2) = x1 + x2 + 8
            filter_2!_A(x1,x2) = x2 + 22
  
    partial status:
  
      pi(if_rm_3#) = [3]
      pi(false_0) = []
      pi(add_2) = []
      pi(rm_2#) = [1, 2]
      pi(eq_2) = [1, 2]
      pi(true_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(rm_2) = []
      pi(nil_0) = []
      pi(minsort_2) = []
      pi(filter_2) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r35, r36, r37, r44, r48, r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    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 + 23
          le_2_A(x1,x2) = 22
          |0|_0_A() = 22
          true_0_A() = 6
          false_0_A() = 21
          rm_2_A(x1,x2) = 2
          nil_0_A() = 1
          minsort_2_A(x1,x2) = x1 + x2 + 7
          filter_2_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        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 + 3
            le_2_A(x1,x2) = x1 + x2 + 7
            |0|_0_A() = 1
            true_0_A() = 8
            false_0_A() = 10
            rm_2_A(x1,x2) = x2 + 8
            nil_0_A() = 7
            minsort_2_A(x1,x2) = x2 + 5
            filter_2_A(x1,x2) = x2 + 10
            eq_2#!_A(x1,x2) = x1 + x2
            s_1!_A(x1) = x1 + 21
            le_2!_A(x1,x2) = 15
            |0|_0!_A() = 18
            true_0!_A() = 19
            false_0!_A() = 20
            rm_2!_A(x1,x2) = 19
            nil_0!_A() = 18
            minsort_2!_A(x1,x2) = x2 + 13
            filter_2!_A(x1,x2) = x2 + 15
  
    partial status:
  
      pi(eq_2#) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(rm_2) = []
      pi(nil_0) = []
      pi(minsort_2) = [1]
      pi(filter_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r28, r38, r39, r44, r48, r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_min_2#_A(x1,x2) = 4
          true_0_A() = 8
          add_2_A(x1,x2) = 2
          min_1#_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 3
          false_0_A() = 9
          |app'|_A(x1,x2) = x2 + 13
          rm_0_A() = 2
          rm_1_A(x1) = 1
          app_2_A(x1,x2) = x1 + x2 + 14
          nil_0_A() = 1
          rm_2_A(x1,x2) = x1 + x2 + 15
          minsort_2_A(x1,x2) = x1 + 14
          filter_2_A(x1,x2) = x2 + 10
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            if_min_2#_A(x1,x2) = x2
            true_0_A() = 6
            add_2_A(x1,x2) = x2 + 14
            min_1#_A(x1) = x1 + 14
            le_2_A(x1,x2) = x1 + x2 + 3
            false_0_A() = 8
            |app'|_A(x1,x2) = x1 + x2
            rm_0_A() = 1
            rm_1_A(x1) = x1 + 1
            app_2_A(x1,x2) = x1 + x2 + 3
            nil_0_A() = 22
            rm_2_A(x1,x2) = x1 + x2 + 16
            minsort_2_A(x1,x2) = x1 + x2 + 3
            filter_2_A(x1,x2) = x1 + x2 + 14
            if_min_2#!_A(x1,x2) = x2
            true_0!_A() = 25
            add_2!_A(x1,x2) = x2 + 12
            min_1#!_A(x1) = x1 + 1
            le_2!_A(x1,x2) = x1 + x2 + 24
            false_0!_A() = 27
            |app'|!_A(x1,x2) = 17
            rm_0!_A() = 1
            rm_1!_A(x1) = 0
            app_2!_A(x1,x2) = x1 + 18
            nil_0!_A() = 22
            rm_2!_A(x1,x2) = 21
            minsort_2!_A(x1,x2) = x1 + x2 + 19
            filter_2!_A(x1,x2) = x2 + 1
  
    partial status:
  
      pi(if_min_2#) = []
      pi(true_0) = []
      pi(add_2) = []
      pi(min_1#) = []
      pi(le_2) = []
      pi(false_0) = []
      pi(|app'|) = [2]
      pi(rm_0) = []
      pi(rm_1) = []
      pi(app_2) = [2]
      pi(nil_0) = []
      pi(rm_2) = [1, 2]
      pi(minsort_2) = []
      pi(filter_2) = [2]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(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

  r35, r36, r37, r44, r48, r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          le_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 7
          le_2_A(x1,x2) = 4
          |0|_0_A() = 2
          true_0_A() = 4
          false_0_A() = 1
          rm_2_A(x1,x2) = x1 + 2
          nil_0_A() = 1
          minsort_2_A(x1,x2) = x1 + 12
          filter_2_A(x1,x2) = x1 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            le_2#_A(x1,x2) = x1 + x2
            s_1_A(x1) = x1 + 3
            le_2_A(x1,x2) = x1 + x2 + 7
            |0|_0_A() = 1
            true_0_A() = 8
            false_0_A() = 10
            rm_2_A(x1,x2) = x2 + 8
            nil_0_A() = 12
            minsort_2_A(x1,x2) = x2 + 5
            filter_2_A(x1,x2) = x2 + 10
            le_2#!_A(x1,x2) = x1 + x2
            s_1!_A(x1) = x1 + 27
            le_2!_A(x1,x2) = 24
            |0|_0!_A() = 1
            true_0!_A() = 6
            false_0!_A() = 0
            rm_2!_A(x1,x2) = 13
            nil_0!_A() = 14
            minsort_2!_A(x1,x2) = x2 + 13
            filter_2!_A(x1,x2) = x2 + 28
  
    partial status:
  
      pi(le_2#) = [2]
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(rm_2) = []
      pi(nil_0) = []
      pi(minsort_2) = []
      pi(filter_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.