YES
We show the termination of the TRS R:

  app(app(minus(),|0|()),y) -> |0|()
  app(app(minus(),app(s(),x)),|0|()) -> app(s(),x)
  app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),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(if(),true()),x),y) -> x
  app(app(app(if(),false()),x),y) -> y
  app(perfectp(),|0|()) -> false()
  app(perfectp(),app(s(),x)) -> app(app(app(app(f(),x),app(s(),|0|())),app(s(),x)),app(s(),x))
  app(app(app(app(f(),|0|()),y),|0|()),u) -> true()
  app(app(app(app(f(),|0|()),y),app(s(),z)),u) -> false()
  app(app(app(app(f(),app(s(),x)),|0|()),z),u) -> app(app(app(app(f(),x),u),app(app(minus(),z),app(s(),x))),u)
  app(app(app(app(f(),app(s(),x)),app(s(),y)),z),u) -> app(app(app(if(),app(app(le(),x),y)),app(app(app(app(f(),app(s(),x)),app(app(minus(),y),x)),z),u)),app(app(app(app(f(),x),u),z),u))
  app(app(map(),|fun|),nil()) -> nil()
  app(app(map(),|fun|),app(app(cons(),x),xs)) -> app(app(cons(),app(|fun|,x)),app(app(map(),|fun|),xs))
  app(app(filter(),|fun|),nil()) -> nil()
  app(app(filter(),|fun|),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(|fun|,x)),|fun|),x),xs)
  app(app(app(app(filter2(),true()),|fun|),x),xs) -> app(app(cons(),x),app(app(filter(),|fun|),xs))
  app(app(app(app(filter2(),false()),|fun|),x),xs) -> app(app(filter(),|fun|),xs)

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(f_0(),y) -> f_1(y)
  app(f_1(x1),y) -> f_2(x1,y)
  app(f_2(x1,x2),y) -> f_3(x1,x2,y)
  app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
  app(if_1(x1),y) -> if_2(x1,y)
  app(if_2(x1,x2),y) -> if_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(minus_0(),y) -> minus_1(y)
  app(minus_1(x1),y) -> minus_2(x1,y)
  app(perfectp_0(),y) -> perfectp_1(y)
  app(s_0(),y) -> s_1(y)
  minus_2(|0|_0(),y) -> |0|_0()
  minus_2(s_1(x),|0|_0()) -> s_1(x)
  minus_2(s_1(x),s_1(y)) -> minus_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)
  if_3(true_0(),x,y) -> x
  if_3(false_0(),x,y) -> y
  perfectp_1(|0|_0()) -> false_0()
  perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
  f_4(|0|_0(),y,|0|_0(),u) -> true_0()
  f_4(|0|_0(),y,s_1(z),u) -> false_0()
  f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
  f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
  map_2(|fun|,nil_0()) -> nil_0()
  map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
  filter_2(|fun|,nil_0()) -> nil_0()
  filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
  filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
  filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)


-- SCC decomposition.

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

p1: app#(f_3(x1,x2,x3),y) -> f_4#(x1,x2,x3,y)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: app#(if_2(x1,x2),y) -> if_3#(x1,x2,y)
p5: app#(le_1(x1),y) -> le_2#(x1,y)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: app#(minus_1(x1),y) -> minus_2#(x1,y)
p8: app#(perfectp_0(),y) -> perfectp_1#(y)
p9: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p10: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)
p11: perfectp_1#(s_1(x)) -> f_4#(x,s_1(|0|_0()),s_1(x),s_1(x))
p12: f_4#(s_1(x),|0|_0(),z,u) -> f_4#(x,u,minus_2(z,s_1(x)),u)
p13: f_4#(s_1(x),|0|_0(),z,u) -> minus_2#(z,s_1(x))
p14: f_4#(s_1(x),s_1(y),z,u) -> if_3#(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
p15: f_4#(s_1(x),s_1(y),z,u) -> le_2#(x,y)
p16: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),z,u)
p17: f_4#(s_1(x),s_1(y),z,u) -> minus_2#(y,x)
p18: f_4#(s_1(x),s_1(y),z,u) -> f_4#(x,u,z,u)
p19: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p20: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p21: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p22: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p23: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p24: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p6, p19, p20, p21, p22, p23, p24}
  {p12, p16, p18}
  {p10}
  {p9}


-- Reduction pair.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p6: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p7: app#(filter_1(x1),y) -> filter_2#(x1,y)
p8: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p9: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r15, r17, r27, r28, r29, r30, r31, r35

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 - 5}
            cons_2_A(x1,x2) = x2 + 119
            app#_A(x1,x2) = max{1, x1 - 17}
            map_1_A(x1) = max{16, x1 + 12}
            filter2_3_A(x1,x2,x3) = max{16, x2 + 12, x3 + 13}
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 5}
            false_0_A = 7
            filter_2#_A(x1,x2) = max{1, x1 - 5}
            filter_1_A(x1) = max{16, x1 + 13}
            app_A(x1,x2) = max{x1 - 133, x2 + 11}
            true_0_A = 11
            if_2_A(x1,x2) = max{x1 - 1, x2 + 143}
            if_3_A(x1,x2,x3) = max{11, x1 - 134, x2, x3}
            le_1_A(x1) = 14
            le_2_A(x1,x2) = 11
            |0|_0_A = 87
            s_1_A(x1) = max{19, x1 - 134}
            f_4_A(x1,x2,x3,x4) = max{7, x1 - 143, x2 - 143, x3 - 143, x4 - 143}
            map_2#!_A(x1,x2) = 114
            cons_2!_A(x1,x2) = max{93, x1 + 85, x2 + 91}
            app#!_A(x1,x2) = 114
            map_1!_A(x1) = 113
            filter2_3!_A(x1,x2,x3) = max{x1 - 12, x2 + 116, x3 + 116}
            filter2_4#!_A(x1,x2,x3,x4) = max{115, x2 + 2}
            false_0!_A = 222
            filter_2#!_A(x1,x2) = max{115, x1 + 2}
            filter_1!_A(x1) = max{116, x1 + 3}
            app!_A(x1,x2) = max{0, x1 - 15}
            true_0!_A = 224
            if_2!_A(x1,x2) = max{3, x1 - 135, x2 - 1}
            if_3!_A(x1,x2,x3) = max{x1 - 134, x2 + 127}
            le_1!_A(x1) = max{225, x1 - 1}
            le_2!_A(x1,x2) = 225
            |0|_0!_A = 223
            s_1!_A(x1) = 226
            f_4!_A(x1,x2,x3,x4) = max{x1 - 106, x2 + 202, x3 - 17, x4 + 135}
        
        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) = 0
              cons_2_A(x1,x2) = 10
              app#_A(x1,x2) = 0
              map_1_A(x1) = x1 + 5
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
              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) = x1 + x2 + 1
              true_0_A() = 12
              if_2_A(x1,x2) = x1 + x2 + 5
              if_3_A(x1,x2,x3) = x2 + x3
              le_1_A(x1) = x1 + 11
              le_2_A(x1,x2) = x2 + 12
              |0|_0_A() = 1
              s_1_A(x1) = x1 + 3
              f_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 9
              map_2#!_A(x1,x2) = x1 + 5
              cons_2!_A(x1,x2) = x1 + 6
              app#!_A(x1,x2) = x1 + 1
              map_1!_A(x1) = 4
              filter2_3!_A(x1,x2,x3) = 0
              filter2_4#!_A(x1,x2,x3,x4) = x2 + 9
              false_0!_A() = 0
              filter_2#!_A(x1,x2) = x1 + 9
              filter_1!_A(x1) = x1 + 10
              app!_A(x1,x2) = 8
              true_0!_A() = 2
              if_2!_A(x1,x2) = 6
              if_3!_A(x1,x2,x3) = 7
              le_1!_A(x1) = 3
              le_2!_A(x1,x2) = 3
              |0|_0!_A() = 1
              s_1!_A(x1) = 4
              f_4!_A(x1,x2,x3,x4) = x1 + x2 + x3 + 5
        
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = [2]
      pi(app#) = [1]
      pi(map_1) = [1]
      pi(filter2_3) = [2]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = [1]
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(if_2) = [1, 2]
      pi(if_3) = [1, 2, 3]
      pi(le_1) = [1]
      pi(le_2) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(f_4) = [1, 2]

The following rules are removed:

  p1, p2, p4, p6, p7



-- SCC decomposition.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p2: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1}
  {p2, p3, p4}


-- Reduction pair.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24

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 - 2}
            cons_2_A(x1,x2) = max{96, x1 - 1, x2 + 1}
            minus_2_A(x1,x2) = max{71, x1 - 72, x2}
            |0|_0_A = 0
            map_2#!_A(x1,x2) = 67
            cons_2!_A(x1,x2) = max{67, x1 - 7, x2 - 100}
            minus_2!_A(x1,x2) = max{0, x1 - 72}
            |0|_0!_A = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x2
              cons_2_A(x1,x2) = x1 + x2 + 5
              minus_2_A(x1,x2) = x1 + 5
              |0|_0_A() = 1
              map_2#!_A(x1,x2) = x2 + 4
              cons_2!_A(x1,x2) = x1 + x2 + 4
              minus_2!_A(x1,x2) = 8
              |0|_0!_A() = 7
        
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(minus_2) = []
      pi(|0|_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p3: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24

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:
          
            filter2_4#_A(x1,x2,x3,x4) = max{22, x2 - 1, x3 - 1, x4 + 13}
            false_0_A = 12
            filter_2#_A(x1,x2) = max{22, x1 - 1, x2 + 13}
            cons_2_A(x1,x2) = max{x1 - 13, x2 + 8}
            app_A(x1,x2) = max{9, x1 - 11, x2 + 8}
            true_0_A = 12
            minus_2_A(x1,x2) = max{46, x1}
            |0|_0_A = 35
            filter2_4#!_A(x1,x2,x3,x4) = max{x3 + 1, x4 + 53}
            false_0!_A = 0
            filter_2#!_A(x1,x2) = x2 + 52
            cons_2!_A(x1,x2) = 0
            app!_A(x1,x2) = max{60, x2 + 39}
            true_0!_A = 4
            minus_2!_A(x1,x2) = max{37, x1 - 36, x2 - 36}
            |0|_0!_A = 27
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              filter2_4#_A(x1,x2,x3,x4) = 0
              false_0_A() = 31
              filter_2#_A(x1,x2) = 0
              cons_2_A(x1,x2) = x2 + 2
              app_A(x1,x2) = x1 + 1
              true_0_A() = 17
              minus_2_A(x1,x2) = x1 + 15
              |0|_0_A() = 1
              filter2_4#!_A(x1,x2,x3,x4) = x2 + x4 + 10
              false_0!_A() = 8
              filter_2#!_A(x1,x2) = x1 + x2 + 9
              cons_2!_A(x1,x2) = 5
              app!_A(x1,x2) = x1 + 1
              true_0!_A() = 8
              minus_2!_A(x1,x2) = 13
              |0|_0!_A() = 10
        
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(minus_2) = []
      pi(|0|_0) = []

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: f_4#(s_1(x),|0|_0(),z,u) -> f_4#(x,u,minus_2(z,s_1(x)),u)
p2: f_4#(s_1(x),s_1(y),z,u) -> f_4#(x,u,z,u)
p3: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),z,u)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24, r25, r26

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:
          
            f_4#_A(x1,x2,x3,x4) = max{0, x1 - 239, x2 - 160, x3 - 81, x4 - 108}
            s_1_A(x1) = max{132, x1 + 108}
            |0|_0_A = 52
            minus_2_A(x1,x2) = max{80, x1, x2 - 158}
            f_4#!_A(x1,x2,x3,x4) = max{103, x1 - 109, x2 - 29, x3 - 82, x4 + 21}
            s_1!_A(x1) = 22
            |0|_0!_A = 104
            minus_2!_A(x1,x2) = 22
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              f_4#_A(x1,x2,x3,x4) = x1 + x3
              s_1_A(x1) = x1 + 6
              |0|_0_A() = 1
              minus_2_A(x1,x2) = x1 + 5
              f_4#!_A(x1,x2,x3,x4) = x1 + x3 + 14
              s_1!_A(x1) = x1 + 18
              |0|_0!_A() = 16
              minus_2!_A(x1,x2) = x2 + 15
        
  
    partial status:
  
      pi(f_4#) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(minus_2) = [1, 2]

The following rules are removed:

  p1, p2



-- SCC decomposition.

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

p1: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),z,u)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),z,u)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24, r25, r26

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:
          
            f_4#_A(x1,x2,x3,x4) = max{0, x1 - 48, x2 - 45, x3 - 4, x4 - 4}
            s_1_A(x1) = max{44, x1 + 32}
            minus_2_A(x1,x2) = max{42, x1 + 32, x2 - 45}
            |0|_0_A = 27
            f_4#!_A(x1,x2,x3,x4) = 0
            s_1!_A(x1) = 90
            minus_2!_A(x1,x2) = 89
            |0|_0!_A = 91
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              f_4#_A(x1,x2,x3,x4) = x2
              s_1_A(x1) = x1 + 6
              minus_2_A(x1,x2) = x1 + 5
              |0|_0_A() = 1
              f_4#!_A(x1,x2,x3,x4) = 11
              s_1!_A(x1) = 13
              minus_2!_A(x1,x2) = 12
              |0|_0!_A() = 11
        
  
    partial status:
  
      pi(f_4#) = [2]
      pi(s_1) = [1]
      pi(minus_2) = [1]
      pi(|0|_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24

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) = 0
            s_1_A(x1) = max{0, x1 - 3}
            minus_2_A(x1,x2) = max{x1, x2}
            |0|_0_A = 0
            le_2#!_A(x1,x2) = 0
            s_1!_A(x1) = 20
            minus_2!_A(x1,x2) = max{20, x1 + 2, x2 + 3}
            |0|_0!_A = 13
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              le_2#_A(x1,x2) = x1 + x2
              s_1_A(x1) = x1 + 10
              minus_2_A(x1,x2) = x1 + x2 + 5
              |0|_0_A() = 6
              le_2#!_A(x1,x2) = x1 + x2
              s_1!_A(x1) = 44
              minus_2!_A(x1,x2) = x1 + 33
              |0|_0!_A() = 26
        
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = []
      pi(minus_2) = [1]
      pi(|0|_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,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_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,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:
          
            minus_2#_A(x1,x2) = max{0, x1 - 155, x2 - 155}
            s_1_A(x1) = max{267, x1 + 154}
            minus_2#!_A(x1,x2) = max{74, x1 - 155}
            s_1!_A(x1) = max{73, x1 - 115}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              minus_2#_A(x1,x2) = x1 + x2
              s_1_A(x1) = x1 + 6
              minus_2#!_A(x1,x2) = x1 + x2 + 13
              s_1!_A(x1) = 12
        
  
    partial status:
  
      pi(minus_2#) = []
      pi(s_1) = []

The following rules are removed:

  p1

Then no dependency pair remains.