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

  r2, r40, r44

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{1, x1 - 1}
            add_2_A(x1,x2) = max{0, x1 - 61}
            |app'|#_A(x1,x2) = max{1, x1 - 2}
            map_1_A(x1) = x1 + 1
            filter2_3_A(x1,x2,x3) = max{x1 - 141, x2 + 1, x3 - 142}
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 2}
            false_0_A = 126
            filter_2#_A(x1,x2) = max{1, x1 - 2}
            filter_1_A(x1) = x1 + 1
            |app'|_A(x1,x2) = max{124, x1 - 48, x2 - 48}
            true_0_A = 128
            add_1_A(x1) = max{0, x1 - 13}
            min_1_A(x1) = max{254, x1 + 61}
            nil_0_A = 127
            rm_2_A(x1,x2) = 259
            map_2#!_A(x1,x2) = x1 + 2
            add_2!_A(x1,x2) = 0
            |app'|#!_A(x1,x2) = 0
            map_1!_A(x1) = x1 + 3
            filter2_3!_A(x1,x2,x3) = max{x1 - 1, x2 + 14, x3 - 130}
            filter2_4#!_A(x1,x2,x3,x4) = 0
            false_0!_A = 5
            filter_2#!_A(x1,x2) = 0
            filter_1!_A(x1) = x1 + 3
            |app'|!_A(x1,x2) = max{x1 + 12, x2 + 9}
            true_0!_A = 3
            add_1!_A(x1) = 0
            min_1!_A(x1) = 0
            nil_0!_A = 0
            rm_2!_A(x1,x2) = 6
        
        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 + 4
              |app'|#_A(x1,x2) = x1 + x2
              map_1_A(x1) = x1 + 12
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 34
              filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 4
              false_0_A() = 4
              filter_2#_A(x1,x2) = x1 + x2
              filter_1_A(x1) = x1 + 12
              |app'|_A(x1,x2) = x1 + x2 + 10
              true_0_A() = 8
              add_1_A(x1) = x1 + 12
              min_1_A(x1) = x1 + 28
              nil_0_A() = 24
              rm_2_A(x1,x2) = x1 + x2 + 35
              map_2#!_A(x1,x2) = 15
              add_2!_A(x1,x2) = x2 + 14
              |app'|#!_A(x1,x2) = x1 + x2
              map_1!_A(x1) = 108
              filter2_3!_A(x1,x2,x3) = 111
              filter2_4#!_A(x1,x2,x3,x4) = x2 + x4 + 15
              false_0!_A() = 11
              filter_2#!_A(x1,x2) = x1 + x2 + 12
              filter_1!_A(x1) = 0
              |app'|!_A(x1,x2) = 109
              true_0!_A() = 13
              add_1!_A(x1) = x1 + 15
              min_1!_A(x1) = x1 + 4
              nil_0!_A() = 12
              rm_2!_A(x1,x2) = x2 + 12
        
  
    partial status:
  
      pi(map_2#) = [1]
      pi(add_2) = [1, 2]
      pi(|app'|#) = [1]
      pi(map_1) = [1]
      pi(filter2_3) = [1, 2, 3]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = [1]
      pi(|app'|) = [1, 2]
      pi(true_0) = []
      pi(add_1) = [1]
      pi(min_1) = [1]
      pi(nil_0) = []
      pi(rm_2) = []

The following rules are removed:

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



-- SCC decomposition.

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 estimated dependency graph contains the following SCCs:

  {p1}


-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{0, x2 - 3}
            add_2_A(x1,x2) = max{11, x1 - 132, x2 + 2}
            map_2#!_A(x1,x2) = max{1, x1 - 1, x2 - 12}
            add_2!_A(x1,x2) = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x2
              add_2_A(x1,x2) = x1 + x2 + 12
              map_2#!_A(x1,x2) = x2 + 9
              add_2!_A(x1,x2) = x2 + 8
        
  
    partial status:
  
      pi(map_2#) = []
      pi(add_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

  r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minsort_2#_A(x1,x2) = max{49, x1 + 36, x2 + 36}
            add_2_A(x1,x2) = max{13, x1 + 9, x2}
            if_minsort_3#_A(x1,x2,x3) = max{37, x1 + 3, x2 + 36, x3 + 36}
            eq_2_A(x1,x2) = 14
            min_1_A(x1) = max{21, x1 + 8}
            false_0_A = 14
            true_0_A = 12
            app_2_A(x1,x2) = max{11, x1, x2}
            rm_2_A(x1,x2) = max{6, x2}
            nil_0_A = 13
            |0|_0_A = 3
            s_1_A(x1) = x1 + 39
            le_2_A(x1,x2) = 14
            if_min_2_A(x1,x2) = max{21, x1 - 31, x2 + 8}
            if_rm_3_A(x1,x2,x3) = max{5, x3}
            minsort_2#!_A(x1,x2) = max{x1 + 8, x2 + 8}
            add_2!_A(x1,x2) = max{49, x1 + 17, x2 + 22}
            if_minsort_3#!_A(x1,x2,x3) = max{18, x1 + 6, x2 + 8, x3 + 8}
            eq_2!_A(x1,x2) = 59
            min_1!_A(x1) = max{36, x1 + 2}
            false_0!_A = 57
            true_0!_A = 62
            app_2!_A(x1,x2) = max{x1 + 35, x2 + 52}
            rm_2!_A(x1,x2) = x2 + 51
            nil_0!_A = 64
            |0|_0!_A = 63
            s_1!_A(x1) = max{58, x1 + 52}
            le_2!_A(x1,x2) = max{x1 + 17, x2 + 53}
            if_min_2!_A(x1,x2) = max{16, x1 - 13}
            if_rm_3!_A(x1,x2,x3) = x3 + 51
        
        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) = 11
              add_2_A(x1,x2) = x1 + x2 + 7
              if_minsort_3#_A(x1,x2,x3) = x1
              eq_2_A(x1,x2) = 11
              min_1_A(x1) = x1 + 2
              false_0_A() = 11
              true_0_A() = 11
              app_2_A(x1,x2) = x1 + x2 + 1
              rm_2_A(x1,x2) = x2
              nil_0_A() = 5
              |0|_0_A() = 9
              s_1_A(x1) = x1 + 2
              le_2_A(x1,x2) = 11
              if_min_2_A(x1,x2) = x2 + 2
              if_rm_3_A(x1,x2,x3) = x3
              minsort_2#!_A(x1,x2) = x1 + x2 + 3
              add_2!_A(x1,x2) = x1 + x2 + 6
              if_minsort_3#!_A(x1,x2,x3) = x2 + x3 + 3
              eq_2!_A(x1,x2) = 14
              min_1!_A(x1) = 14
              false_0!_A() = 11
              true_0!_A() = 13
              app_2!_A(x1,x2) = x1 + x2 + 5
              rm_2!_A(x1,x2) = x2 + 5
              nil_0!_A() = 5
              |0|_0!_A() = 14
              s_1!_A(x1) = 12
              le_2!_A(x1,x2) = x1 + x2 + 5
              if_min_2!_A(x1,x2) = x1 + 2
              if_rm_3!_A(x1,x2,x3) = x3 + 4
        
  
    partial status:
  
      pi(minsort_2#) = []
      pi(add_2) = [1, 2]
      pi(if_minsort_3#) = []
      pi(eq_2) = [1]
      pi(min_1) = [1]
      pi(false_0) = []
      pi(true_0) = []
      pi(app_2) = [1, 2]
      pi(rm_2) = [2]
      pi(nil_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(le_2) = [1]
      pi(if_min_2) = [2]
      pi(if_rm_3) = [3]

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

  r31, r32, r33, r34

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minsort_2#_A(x1,x2) = max{4, x1 - 15}
            add_2_A(x1,x2) = x2 + 18
            if_minsort_3#_A(x1,x2,x3) = max{4, x1 - 311, x2 - 19}
            eq_2_A(x1,x2) = 310
            min_1_A(x1) = max{0, x1 - 16}
            false_0_A = 280
            |0|_0_A = 126
            true_0_A = 155
            s_1_A(x1) = max{3, x1 + 2}
            minsort_2#!_A(x1,x2) = x1 + 135
            add_2!_A(x1,x2) = 119
            if_minsort_3#!_A(x1,x2,x3) = max{x1 - 158, x2 + 135}
            eq_2!_A(x1,x2) = max{27, x1 - 4}
            min_1!_A(x1) = max{279, x1 + 136}
            false_0!_A = 1
            |0|_0!_A = 122
            true_0!_A = 121
            s_1!_A(x1) = 120
        
        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) = x2 + 9
              if_minsort_3#_A(x1,x2,x3) = x2 + x3
              eq_2_A(x1,x2) = x1 + 13
              min_1_A(x1) = 4
              false_0_A() = 20
              |0|_0_A() = 7
              true_0_A() = 11
              s_1_A(x1) = x1 + 9
              minsort_2#!_A(x1,x2) = x1 + 17
              add_2!_A(x1,x2) = x2 + 19
              if_minsort_3#!_A(x1,x2,x3) = x2 + 9
              eq_2!_A(x1,x2) = 25
              min_1!_A(x1) = x1 + 1
              false_0!_A() = 26
              |0|_0!_A() = 23
              true_0!_A() = 24
              s_1!_A(x1) = 22
        
  
    partial status:
  
      pi(minsort_2#) = []
      pi(add_2) = []
      pi(if_minsort_3#) = []
      pi(eq_2) = []
      pi(min_1) = [1]
      pi(false_0) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app_2#_A(x1,x2) = max{0, x1 - 77}
            add_2_A(x1,x2) = max{126, x1 - 255, x2 + 76}
            app_2#!_A(x1,x2) = max{12, x1 - 1}
            add_2!_A(x1,x2) = x1 + 11
        
        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) = 0
              add_2_A(x1,x2) = x1 + 10
              app_2#!_A(x1,x2) = x1 + 17
              add_2!_A(x1,x2) = x1 + 16
        
  
    partial status:
  
      pi(app_2#) = []
      pi(add_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

  r31, r32, r33, r34

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_rm_3#_A(x1,x2,x3) = max{0, x1 - 30, x2 - 39, x3 - 52}
            false_0_A = 16
            add_2_A(x1,x2) = max{x1 + 23, x2 + 26}
            rm_2#_A(x1,x2) = max{0, x1 - 39, x2 - 52}
            eq_2_A(x1,x2) = max{14, x2 + 1}
            true_0_A = 29
            |0|_0_A = 28
            s_1_A(x1) = x1 + 29
            if_rm_3#!_A(x1,x2,x3) = max{0, x1 - 15, x3 - 28}
            false_0!_A = 96
            add_2!_A(x1,x2) = max{103, x1 - 2}
            rm_2#!_A(x1,x2) = max{0, x2 - 28}
            eq_2!_A(x1,x2) = max{98, x1 - 29}
            true_0!_A = 96
            |0|_0!_A = 97
            s_1!_A(x1) = max{100, x1 - 30}
        
        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() = 14
              add_2_A(x1,x2) = x1 + x2 + 20
              rm_2#_A(x1,x2) = x2 + 20
              eq_2_A(x1,x2) = x1 + 29
              true_0_A() = 10
              |0|_0_A() = 4
              s_1_A(x1) = x1 + 4
              if_rm_3#!_A(x1,x2,x3) = 43
              false_0!_A() = 49
              add_2!_A(x1,x2) = 44
              rm_2#!_A(x1,x2) = 45
              eq_2!_A(x1,x2) = x1 + 46
              true_0!_A() = 42
              |0|_0!_A() = 43
              s_1!_A(x1) = 1
        
  
    partial status:
  
      pi(if_rm_3#) = []
      pi(false_0) = []
      pi(add_2) = []
      pi(rm_2#) = []
      pi(eq_2) = []
      pi(true_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r54

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            eq_2#_A(x1,x2) = max{0, x1 - 2, x2 - 2}
            s_1_A(x1) = max{3, x1 + 1}
            filter_2_A(x1,x2) = max{30, x1 - 31, x2 - 31}
            nil_0_A = 5
            eq_2#!_A(x1,x2) = max{0, x2 - 4}
            s_1!_A(x1) = x1 + 27
            filter_2!_A(x1,x2) = max{x1 + 21, x2 - 6}
            nil_0!_A = 20
        
        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
              filter_2_A(x1,x2) = x2 + 10
              nil_0_A() = 9
              eq_2#!_A(x1,x2) = x2 + 14
              s_1!_A(x1) = 13
              filter_2!_A(x1,x2) = x2 + 15
              nil_0!_A() = 11
        
  
    partial status:
  
      pi(eq_2#) = [2]
      pi(s_1) = [1]
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r35, r36, r37

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_min_2#_A(x1,x2) = max{619, x1 + 85, x2 + 588}
            true_0_A = 342
            add_2_A(x1,x2) = max{x1 + 10, x2 + 15}
            min_1#_A(x1) = max{619, x1 + 588}
            le_2_A(x1,x2) = max{344, x1 + 325}
            false_0_A = 511
            |0|_0_A = 85
            s_1_A(x1) = max{186, x1 + 83}
            if_min_2#!_A(x1,x2) = max{1, x1 - 322, x2 - 7}
            true_0!_A = 172
            add_2!_A(x1,x2) = max{188, x1 + 18, x2 + 160}
            min_1#!_A(x1) = max{21, x1 - 7}
            le_2!_A(x1,x2) = 188
            false_0!_A = 187
            |0|_0!_A = 188
            s_1!_A(x1) = max{189, x1 + 1}
        
        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) = 0
              true_0_A() = 5
              add_2_A(x1,x2) = x1 + x2 + 4
              min_1#_A(x1) = 0
              le_2_A(x1,x2) = x1 + x2 + 3
              false_0_A() = 7
              |0|_0_A() = 2
              s_1_A(x1) = x1 + 4
              if_min_2#!_A(x1,x2) = x2
              true_0!_A() = 10
              add_2!_A(x1,x2) = x1 + x2 + 7
              min_1#!_A(x1) = x1 + 2
              le_2!_A(x1,x2) = x1 + 9
              false_0!_A() = 12
              |0|_0!_A() = 13
              s_1!_A(x1) = 11
        
  
    partial status:
  
      pi(if_min_2#) = []
      pi(true_0) = []
      pi(add_2) = [1]
      pi(min_1#) = []
      pi(le_2) = []
      pi(false_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le_2#_A(x1,x2) = max{0, x1 - 10, x2 - 10}
            s_1_A(x1) = max{10, x1 + 9}
            le_2#!_A(x1,x2) = max{5, x1 - 3, x2 - 1}
            s_1!_A(x1) = max{4, x1 - 2}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              le_2#_A(x1,x2) = 0
              s_1_A(x1) = 3
              le_2#!_A(x1,x2) = x1 + x2 + 10
              s_1!_A(x1) = 9
        
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.