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

  r35, r36, r37

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x2
              add_2_A(x1,x2) = x1 + x2 + 5
              |app'|#_A(x1,x2) = x2
              map_1_A(x1) = x1
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3
              filter2_4#_A(x1,x2,x3,x4) = x4
              false_0_A() = 34
              filter_2#_A(x1,x2) = x2
              filter_1_A(x1) = x1
              |app'|_A(x1,x2) = x1 + x2 + 1
              true_0_A() = 56
              le_2_A(x1,x2) = x1 + x2 + 31
              |0|_0_A() = 25
              s_1_A(x1) = x1 + 3
              map_2#!_A(x1,x2) = 16
              add_2!_A(x1,x2) = x2 + 17
              |app'|#!_A(x1,x2) = 16
              map_1!_A(x1) = 16
              filter2_3!_A(x1,x2,x3) = 22
              filter2_4#!_A(x1,x2,x3,x4) = 15
              false_0!_A() = 13
              filter_2#!_A(x1,x2) = 14
              filter_1!_A(x1) = x1 + 15
              |app'|!_A(x1,x2) = 21
              true_0!_A() = 44
              le_2!_A(x1,x2) = x1 + x2 + 20
              |0|_0!_A() = 45
              s_1!_A(x1) = 47
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{1, x1 - 45}
            add_2_A(x1,x2) = x2 + 43
            |app'|#_A(x1,x2) = max{1, x1 - 81}
            map_1_A(x1) = x1 + 36
            filter2_3_A(x1,x2,x3) = max{x1 - 19, x2 + 21, x3 - 1}
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 60}
            false_0_A = 17
            filter_2#_A(x1,x2) = max{1, x1 - 60}
            filter_1_A(x1) = x1 + 80
            |app'|_A(x1,x2) = max{19, x1 - 81, x2 - 81}
            true_0_A = 15
            le_2_A(x1,x2) = max{99, x1 - 21, x2 + 14}
            |0|_0_A = 2
            s_1_A(x1) = max{20, x1 + 1}
            map_2#!_A(x1,x2) = x1 + 14
            add_2!_A(x1,x2) = max{x1 + 12, x2 + 5}
            |app'|#!_A(x1,x2) = max{0, x1 - 21}
            map_1!_A(x1) = 0
            filter2_3!_A(x1,x2,x3) = max{0, x3 - 20}
            filter2_4#!_A(x1,x2,x3,x4) = max{0, x3 - 21, x4 - 1}
            false_0!_A = 436
            filter_2#!_A(x1,x2) = x1 + 58
            filter_1!_A(x1) = x1 + 59
            |app'|!_A(x1,x2) = max{25, x1 + 2}
            true_0!_A = 1
            le_2!_A(x1,x2) = 4
            |0|_0!_A = 437
            s_1!_A(x1) = 0
        
  
    partial status:
  
      pi(map_2#) = [2]
      pi(add_2) = [1, 2]
      pi(|app'|#) = [2]
      pi(map_1) = [1]
      pi(filter2_3) = [2]
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = [1]
      pi(|app'|) = [1, 2]
      pi(true_0) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(s_1) = [1]

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

  r31, r32, r33, r34, r35, r36, r37, r38, r39, r44, r45, r46, r47, r48, 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!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              minsort_2#_A(x1,x2) = 0
              add_2_A(x1,x2) = x2 + 21
              if_minsort_3#_A(x1,x2,x3) = 0
              eq_2_A(x1,x2) = 1
              min_1_A(x1) = x1 + 3
              false_0_A() = 1
              true_0_A() = 1
              app_2_A(x1,x2) = x1 + x2 + 5
              rm_2_A(x1,x2) = x2
              nil_0_A() = 6
              |0|_0_A() = 1
              s_1_A(x1) = x1 + 9
              le_2_A(x1,x2) = 11
              if_rm_3_A(x1,x2,x3) = x3
              minsort_2_A(x1,x2) = x1 + x2 + 5
              filter_2_A(x1,x2) = x1 + 10
              minsort_2#!_A(x1,x2) = x1 + x2 + 13
              add_2!_A(x1,x2) = x2 + 23
              if_minsort_3#!_A(x1,x2,x3) = x1 + x2 + x3 + 12
              eq_2!_A(x1,x2) = 41
              min_1!_A(x1) = x1 + 1
              false_0!_A() = 42
              true_0!_A() = 40
              app_2!_A(x1,x2) = x1 + x2 + 8
              rm_2!_A(x1,x2) = x2 + 2
              nil_0!_A() = 7
              |0|_0!_A() = 43
              s_1!_A(x1) = 43
              le_2!_A(x1,x2) = 42
              if_rm_3!_A(x1,x2,x3) = x3 + 2
              minsort_2!_A(x1,x2) = x1 + 43
              filter_2!_A(x1,x2) = x2 + 41
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minsort_2#_A(x1,x2) = max{39, x2}
            add_2_A(x1,x2) = 0
            if_minsort_3#_A(x1,x2,x3) = max{39, x1 - 2, x2}
            eq_2_A(x1,x2) = 40
            min_1_A(x1) = max{77, x1 + 37}
            false_0_A = 40
            true_0_A = 21
            app_2_A(x1,x2) = max{x1 + 12, x2 + 53}
            rm_2_A(x1,x2) = max{77, x1 - 12}
            nil_0_A = 0
            |0|_0_A = 25
            s_1_A(x1) = max{44, x1 + 23}
            le_2_A(x1,x2) = max{78, x1 - 5}
            if_rm_3_A(x1,x2,x3) = max{77, x2 - 12, x3 + 12}
            minsort_2_A(x1,x2) = max{79, x1 + 26, x2 - 16}
            filter_2_A(x1,x2) = max{14, x2 - 1}
            minsort_2#!_A(x1,x2) = x2 + 25
            add_2!_A(x1,x2) = 0
            if_minsort_3#!_A(x1,x2,x3) = max{x1 - 15, x2 + 8}
            eq_2!_A(x1,x2) = max{4, x1 + 2}
            min_1!_A(x1) = x1 + 24
            false_0!_A = 26
            true_0!_A = 28
            app_2!_A(x1,x2) = max{x1 + 12, x2 + 27}
            rm_2!_A(x1,x2) = 0
            nil_0!_A = 7
            |0|_0!_A = 27
            s_1!_A(x1) = 2
            le_2!_A(x1,x2) = max{29, x1 - 45, x2}
            if_rm_3!_A(x1,x2,x3) = max{0, x3 - 1}
            minsort_2!_A(x1,x2) = max{29, x2 + 12}
            filter_2!_A(x1,x2) = max{0, x1 - 73}
        
  
    partial status:
  
      pi(minsort_2#) = []
      pi(add_2) = []
      pi(if_minsort_3#) = []
      pi(eq_2) = [1]
      pi(min_1) = [1]
      pi(false_0) = []
      pi(true_0) = []
      pi(app_2) = [2]
      pi(rm_2) = []
      pi(nil_0) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(le_2) = []
      pi(if_rm_3) = []
      pi(minsort_2) = [1, 2]
      pi(filter_2) = [1]

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, r35, r36, r37, 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!):
        
          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 + 3
              if_minsort_3#_A(x1,x2,x3) = x2 + x3
              eq_2_A(x1,x2) = x1 + 2
              min_1_A(x1) = 3
              false_0_A() = 1
              |0|_0_A() = 5
              true_0_A() = 6
              s_1_A(x1) = x1 + 5
              le_2_A(x1,x2) = x1 + 2
              rm_2_A(x1,x2) = x1 + x2 + 3
              nil_0_A() = 3
              minsort_2#!_A(x1,x2) = x1 + 4
              add_2!_A(x1,x2) = 6
              if_minsort_3#!_A(x1,x2,x3) = x2 + 2
              eq_2!_A(x1,x2) = x1 + 5
              min_1!_A(x1) = 5
              false_0!_A() = 9
              |0|_0!_A() = 10
              true_0!_A() = 7
              s_1!_A(x1) = 13
              le_2!_A(x1,x2) = x2 + 8
              rm_2!_A(x1,x2) = 6
              nil_0!_A() = 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minsort_2#_A(x1,x2) = max{2, x1 - 21, x2 - 39}
            add_2_A(x1,x2) = max{22, x1 - 70, x2}
            if_minsort_3#_A(x1,x2,x3) = max{2, x1 - 19, x2 - 21, x3 - 39}
            eq_2_A(x1,x2) = max{8, x1 - 118}
            min_1_A(x1) = max{20, x1 - 2}
            false_0_A = 1
            |0|_0_A = 117
            true_0_A = 1
            s_1_A(x1) = max{8, x1 + 1}
            le_2_A(x1,x2) = max{8, x1 - 118}
            rm_2_A(x1,x2) = max{54, x1 - 24, x2 - 24}
            nil_0_A = 10
            minsort_2#!_A(x1,x2) = max{0, x1 - 25, x2 - 2}
            add_2!_A(x1,x2) = max{19, x1 + 17}
            if_minsort_3#!_A(x1,x2,x3) = max{0, x1 - 9, x2 - 25}
            eq_2!_A(x1,x2) = max{89, x1 - 96}
            min_1!_A(x1) = max{0, x1 - 53}
            false_0!_A = 14
            |0|_0!_A = 21
            true_0!_A = 20
            s_1!_A(x1) = max{15, x1 - 125}
            le_2!_A(x1,x2) = max{89, x2 - 124}
            rm_2!_A(x1,x2) = max{13, x1 - 97, x2 - 23}
            nil_0!_A = 11
        
  
    partial status:
  
      pi(minsort_2#) = []
      pi(add_2) = []
      pi(if_minsort_3#) = []
      pi(eq_2) = []
      pi(min_1) = []
      pi(false_0) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(le_2) = []
      pi(rm_2) = []
      pi(nil_0) = []

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

  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!):
        
          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 + 5
              rm_2_A(x1,x2) = x1 + x2 + 8
              nil_0_A() = 6
              app_2#!_A(x1,x2) = 0
              add_2!_A(x1,x2) = 7
              rm_2!_A(x1,x2) = 5
              nil_0!_A() = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app_2#_A(x1,x2) = 0
            add_2_A(x1,x2) = 0
            rm_2_A(x1,x2) = max{10, x1 - 44, x2 - 10}
            nil_0_A = 9
            app_2#!_A(x1,x2) = max{x1 + 41, x2 + 41}
            add_2!_A(x1,x2) = max{39, x1 - 65, x2 - 76}
            rm_2!_A(x1,x2) = max{44, x1 - 66, x2 - 10}
            nil_0!_A = 0
        
  
    partial status:
  
      pi(app_2#) = [1]
      pi(add_2) = [1, 2]
      pi(rm_2) = [1]
      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_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!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              if_rm_3#_A(x1,x2,x3) = 0
              false_0_A() = 11
              add_2_A(x1,x2) = x1
              rm_2#_A(x1,x2) = 0
              eq_2_A(x1,x2) = x2 + 9
              true_0_A() = 11
              |0|_0_A() = 2
              s_1_A(x1) = x1 + 3
              if_rm_3#!_A(x1,x2,x3) = x1
              false_0!_A() = 8
              add_2!_A(x1,x2) = x2 + 11
              rm_2#!_A(x1,x2) = x2 + 10
              eq_2!_A(x1,x2) = 9
              true_0!_A() = 8
              |0|_0!_A() = 9
              s_1!_A(x1) = 10
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_rm_3#_A(x1,x2,x3) = max{1, x1 - 82, x2 - 2}
            false_0_A = 4
            add_2_A(x1,x2) = max{121, x2 - 29}
            rm_2#_A(x1,x2) = max{1, x1 - 2}
            eq_2_A(x1,x2) = max{81, x1 - 9}
            true_0_A = 10
            |0|_0_A = 0
            s_1_A(x1) = x1
            if_rm_3#!_A(x1,x2,x3) = 52
            false_0!_A = 3
            add_2!_A(x1,x2) = max{151, x2 + 54}
            rm_2#!_A(x1,x2) = 53
            eq_2!_A(x1,x2) = max{177, x2 + 4}
            true_0!_A = 176
            |0|_0!_A = 2
            s_1!_A(x1) = max{0, x1 - 1}
        
  
    partial status:
  
      pi(if_rm_3#) = [1, 3]
      pi(false_0) = []
      pi(add_2) = [2]
      pi(rm_2#) = [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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              eq_2#_A(x1,x2) = x1 + x2
              s_1_A(x1) = x1 + 3
              eq_2#!_A(x1,x2) = x1 + x2
              s_1!_A(x1) = 8
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            eq_2#_A(x1,x2) = max{x1, x2}
            s_1_A(x1) = x1
            eq_2#!_A(x1,x2) = max{x1 + 98, x2 + 98}
            s_1!_A(x1) = max{96, x1 - 53}
        
  
    partial status:
  
      pi(eq_2#) = []
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: if_min_2#(true_0(),add_2(n,add_2(m,x))) -> min_1#(add_2(n,x))
p2: min_1#(add_2(n,add_2(m,x))) -> if_min_2#(le_2(n,m),add_2(n,add_2(m,x)))
p3: if_min_2#(false_0(),add_2(n,add_2(m,x))) -> min_1#(add_2(m,x))

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r36, 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!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              if_min_2#_A(x1,x2) = 0
              true_0_A() = 34
              add_2_A(x1,x2) = x1 + 10
              min_1#_A(x1) = 0
              le_2_A(x1,x2) = x2 + 7
              false_0_A() = 21
              s_1_A(x1) = x1 + 1
              |0|_0_A() = 14
              rm_2_A(x1,x2) = x1 + x2 + 12
              nil_0_A() = 1
              if_min_2#!_A(x1,x2) = 2
              true_0!_A() = 0
              add_2!_A(x1,x2) = x1 + x2 + 3
              min_1#!_A(x1) = 2
              le_2!_A(x1,x2) = x2 + 1
              false_0!_A() = 14
              s_1!_A(x1) = 15
              |0|_0!_A() = 15
              rm_2!_A(x1,x2) = x2 + 10
              nil_0!_A() = 4
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            if_min_2#_A(x1,x2) = max{2, x2 - 60}
            true_0_A = 3
            add_2_A(x1,x2) = max{43, x2 + 19}
            min_1#_A(x1) = max{1, x1 - 55}
            le_2_A(x1,x2) = x1 + 52
            false_0_A = 3
            s_1_A(x1) = max{34, x1 + 19}
            |0|_0_A = 34
            rm_2_A(x1,x2) = max{62, x1 - 18, x2 + 32}
            nil_0_A = 93
            if_min_2#!_A(x1,x2) = max{31, x2 - 35}
            true_0!_A = 3
            add_2!_A(x1,x2) = max{x1 + 49, x2 + 28}
            min_1#!_A(x1) = max{2, x1 - 31}
            le_2!_A(x1,x2) = 48
            false_0!_A = 1
            s_1!_A(x1) = 87
            |0|_0!_A = 4
            rm_2!_A(x1,x2) = max{76, x1 - 135, x2 - 103}
            nil_0!_A = 50
        
  
    partial status:
  
      pi(if_min_2#) = []
      pi(true_0) = []
      pi(add_2) = []
      pi(min_1#) = []
      pi(le_2) = [1, 2]
      pi(false_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(rm_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1, p3



-- SCC decomposition.

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

p1: min_1#(add_2(n,add_2(m,x))) -> if_min_2#(le_2(n,m),add_2(n,add_2(m,x)))

and R consists of:

r1: |app'|(add_0(),y) -> add_1(y)
r2: |app'|(add_1(x1),y) -> add_2(x1,y)
r3: |app'|(app_0(),y) -> app_1(y)
r4: |app'|(app_1(x1),y) -> app_2(x1,y)
r5: |app'|(eq_0(),y) -> eq_1(y)
r6: |app'|(eq_1(x1),y) -> eq_2(x1,y)
r7: |app'|(filter_0(),y) -> filter_1(y)
r8: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r9: |app'|(filter2_0(),y) -> filter2_1(y)
r10: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r11: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: |app'|(if_min_0(),y) -> if_min_1(y)
r14: |app'|(if_min_1(x1),y) -> if_min_2(x1,y)
r15: |app'|(if_minsort_0(),y) -> if_minsort_1(y)
r16: |app'|(if_minsort_1(x1),y) -> if_minsort_2(x1,y)
r17: |app'|(if_minsort_2(x1,x2),y) -> if_minsort_3(x1,x2,y)
r18: |app'|(if_rm_0(),y) -> if_rm_1(y)
r19: |app'|(if_rm_1(x1),y) -> if_rm_2(x1,y)
r20: |app'|(if_rm_2(x1,x2),y) -> if_rm_3(x1,x2,y)
r21: |app'|(le_0(),y) -> le_1(y)
r22: |app'|(le_1(x1),y) -> le_2(x1,y)
r23: |app'|(map_0(),y) -> map_1(y)
r24: |app'|(map_1(x1),y) -> map_2(x1,y)
r25: |app'|(min_0(),y) -> min_1(y)
r26: |app'|(minsort_0(),y) -> minsort_1(y)
r27: |app'|(minsort_1(x1),y) -> minsort_2(x1,y)
r28: |app'|(rm_0(),y) -> rm_1(y)
r29: |app'|(rm_1(x1),y) -> rm_2(x1,y)
r30: |app'|(s_0(),y) -> s_1(y)
r31: eq_2(|0|_0(),|0|_0()) -> true_0()
r32: eq_2(|0|_0(),s_1(x)) -> false_0()
r33: eq_2(s_1(x),|0|_0()) -> false_0()
r34: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r35: le_2(|0|_0(),y) -> true_0()
r36: le_2(s_1(x),|0|_0()) -> false_0()
r37: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r38: app_2(nil_0(),y) -> y
r39: app_2(add_2(n,x),y) -> add_2(n,app_2(x,y))
r40: min_1(add_2(n,nil_0())) -> n
r41: min_1(add_2(n,add_2(m,x))) -> if_min_2(le_2(n,m),add_2(n,add_2(m,x)))
r42: if_min_2(true_0(),add_2(n,add_2(m,x))) -> min_1(add_2(n,x))
r43: if_min_2(false_0(),add_2(n,add_2(m,x))) -> min_1(add_2(m,x))
r44: rm_2(n,nil_0()) -> nil_0()
r45: rm_2(n,add_2(m,x)) -> if_rm_3(eq_2(n,m),n,add_2(m,x))
r46: if_rm_3(true_0(),n,add_2(m,x)) -> rm_2(n,x)
r47: if_rm_3(false_0(),n,add_2(m,x)) -> add_2(m,rm_2(n,x))
r48: minsort_2(nil_0(),nil_0()) -> nil_0()
r49: minsort_2(add_2(n,x),y) -> if_minsort_3(eq_2(n,min_1(add_2(n,x))),add_2(n,x),y)
r50: if_minsort_3(true_0(),add_2(n,x),y) -> add_2(n,minsort_2(app_2(rm_2(n,x),y),nil_0()))
r51: if_minsort_3(false_0(),add_2(n,x),y) -> minsort_2(x,add_2(n,y))
r52: map_2(f,nil_0()) -> nil_0()
r53: map_2(f,add_2(x,xs)) -> add_2(|app'|(f,x),map_2(f,xs))
r54: filter_2(f,nil_0()) -> nil_0()
r55: filter_2(f,add_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r56: filter2_4(true_0(),f,x,xs) -> add_2(x,filter_2(f,xs))
r57: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: 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!):
        
          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 + 2
              le_2#!_A(x1,x2) = x1 + x2
              s_1!_A(x1) = 51
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le_2#_A(x1,x2) = 0
            s_1_A(x1) = max{2, x1 - 101}
            le_2#!_A(x1,x2) = x2 + 115
            s_1!_A(x1) = x1 + 116
        
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.