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(or(),true()),y) -> true()
  app(app(or(),false()),y) -> y
  app(app(union(),empty()),h) -> h
  app(app(union(),app(app(app(edge(),x),y),i)),h) -> app(app(app(edge(),x),y),app(app(union(),i),h))
  app(app(app(app(reach(),x),y),empty()),h) -> false()
  app(app(app(app(reach(),x),y),app(app(app(edge(),u),v),i)),h) -> app(app(app(app(app(if_reach_1(),app(app(eq(),x),u)),x),y),app(app(app(edge(),u),v),i)),h)
  app(app(app(app(app(if_reach_1(),true()),x),y),app(app(app(edge(),u),v),i)),h) -> app(app(app(app(app(if_reach_2(),app(app(eq(),y),v)),x),y),app(app(app(edge(),u),v),i)),h)
  app(app(app(app(app(if_reach_1(),false()),x),y),app(app(app(edge(),u),v),i)),h) -> app(app(app(app(reach(),x),y),i),app(app(app(edge(),u),v),h))
  app(app(app(app(app(if_reach_2(),true()),x),y),app(app(app(edge(),u),v),i)),h) -> true()
  app(app(app(app(app(if_reach_2(),false()),x),y),app(app(app(edge(),u),v),i)),h) -> app(app(or(),app(app(app(app(reach(),x),y),i),h)),app(app(app(app(reach(),v),y),app(app(union(),i),h)),empty()))
  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
  app(app(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
  app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),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(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(edge_0(),y) -> edge_1(y)
  app(edge_1(x1),y) -> edge_2(x1,y)
  app(edge_2(x1,x2),y) -> edge_3(x1,x2,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_reach_1_0(),y) -> if_reach_1_1(y)
  app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
  app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
  app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
  app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
  app(if_reach_2_0(),y) -> if_reach_2_1(y)
  app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
  app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
  app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
  app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(or_0(),y) -> or_1(y)
  app(or_1(x1),y) -> or_2(x1,y)
  app(reach_0(),y) -> reach_1(y)
  app(reach_1(x1),y) -> reach_2(x1,y)
  app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
  app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
  app(s_0(),y) -> s_1(y)
  app(union_0(),y) -> union_1(y)
  app(union_1(x1),y) -> union_2(x1,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)
  or_2(true_0(),y) -> true_0()
  or_2(false_0(),y) -> y
  union_2(empty_0(),h) -> h
  union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
  reach_4(x,y,empty_0(),h) -> false_0()
  reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
  if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
  if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
  if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
  if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
  filter2_4(true_0(),f,x,xs) -> cons_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#(eq_1(x1),y) -> eq_2#(x1,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_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5#(x1,x2,x3,x4,y)
p5: app#(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5#(x1,x2,x3,x4,y)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: app#(or_1(x1),y) -> or_2#(x1,y)
p8: app#(reach_3(x1,x2,x3),y) -> reach_4#(x1,x2,x3,y)
p9: app#(union_1(x1),y) -> union_2#(x1,y)
p10: eq_2#(s_1(x),s_1(y)) -> eq_2#(x,y)
p11: union_2#(edge_3(x,y,i),h) -> union_2#(i,h)
p12: reach_4#(x,y,edge_3(u,v,i),h) -> if_reach_1_5#(eq_2(x,u),x,y,edge_3(u,v,i),h)
p13: reach_4#(x,y,edge_3(u,v,i),h) -> eq_2#(x,u)
p14: if_reach_1_5#(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5#(eq_2(y,v),x,y,edge_3(u,v,i),h)
p15: if_reach_1_5#(true_0(),x,y,edge_3(u,v,i),h) -> eq_2#(y,v)
p16: if_reach_1_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,edge_3(u,v,h))
p17: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> or_2#(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
p18: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,h)
p19: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(v,y,union_2(i,h),empty_0())
p20: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> union_2#(i,h)
p21: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p22: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p23: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p24: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p25: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p26: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p6, p21, p22, p23, p24, p25, p26}
  {p12, p14, p16, p18, p19}
  {p10}
  {p11}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_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,cons_2(x,xs)) -> app#(f,x)
p7: app#(filter_1(x1),y) -> filter_2#(x1,y)
p8: filter_2#(f,cons_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(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r10, r11, r12, r14, r15, r16, r17, r19, r20, r21, r22, r24, r26, r27, r28, r29, r30, r32, r33, r35, r36, r37, r38, r39, r40, r43, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x1 + x2 + 4
          cons_2_A(x1,x2) = x1 + x2 + 3
          app#_A(x1,x2) = x1 + x2 + 4
          map_1_A(x1) = x1 + 2
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5
          false_0_A() = 2
          filter_2#_A(x1,x2) = x1 + x2 + 4
          filter_1_A(x1) = x1 + 2
          app_A(x1,x2) = x1 + x2 + 4
          true_0_A() = 3
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          edge_0_A() = 1
          edge_1_A(x1) = x1 + 2
          edge_2_A(x1,x2) = x1 + x2 + 3
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 5
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 2
          eq_2_A(x1,x2) = x1 + x2 + 4
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          if_reach_1_0_A() = 1
          if_reach_1_1_A(x1) = x1 + 2
          if_reach_1_2_A(x1,x2) = x1 + x2 + 3
          if_reach_1_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 1
          if_reach_2_0_A() = 1
          if_reach_2_1_A(x1) = x1 + 2
          if_reach_2_2_A(x1,x2) = x1 + x2 + 3
          if_reach_2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_2_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          map_0_A() = 1
          or_0_A() = 1
          or_1_A(x1) = x1 + 2
          or_2_A(x1,x2) = x1 + x2 + 3
          reach_0_A() = 1
          reach_1_A(x1) = x1 + 2
          reach_2_A(x1,x2) = x1 + x2 + 3
          reach_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          union_0_A() = 1
          union_1_A(x1) = x1 + 2
          |0|_0_A() = 4
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 6
          empty_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app# > filter2_4# > filter_2# = if_reach_1_2 = filter_2 > map_2# = app = if_reach_1_4 > map_1 = filter2_3 = if_reach_1_1 = if_reach_1_3 = if_reach_2_4 = union_1 = reach_4 > cons_2 = filter_1 = cons_0 = eq_2 = filter2_2 = if_reach_1_0 = if_reach_2_3 = map_0 = or_2 = reach_3 = union_0 = nil_0 > cons_1 = edge_3 = eq_1 = filter_0 = filter2_1 = if_reach_2_2 = or_1 = reach_2 = |0|_0 > true_0 = edge_2 = eq_0 = filter2_0 = if_reach_2_1 = or_0 = reach_1 = s_1 > false_0 = edge_1 = if_reach_2_0 = reach_0 = s_0 > edge_0 = empty_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(app#) = [2]
      pi(map_1) = [1]
      pi(filter2_3) = [3]
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = [1]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(edge_0) = []
      pi(edge_1) = [1]
      pi(edge_2) = [1, 2]
      pi(edge_3) = [3]
      pi(eq_0) = []
      pi(eq_1) = [1]
      pi(eq_2) = [1, 2]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = [1]
      pi(if_reach_1_2) = [1, 2]
      pi(if_reach_1_3) = [3]
      pi(if_reach_1_4) = [3]
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = [1]
      pi(if_reach_2_2) = [1, 2]
      pi(if_reach_2_3) = [1, 2, 3]
      pi(if_reach_2_4) = [4]
      pi(map_0) = []
      pi(or_0) = []
      pi(or_1) = [1]
      pi(or_2) = []
      pi(reach_0) = []
      pi(reach_1) = [1]
      pi(reach_2) = [1, 2]
      pi(reach_3) = [3]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(union_0) = []
      pi(union_1) = [1]
      pi(|0|_0) = []
      pi(reach_4) = [1, 3]
      pi(empty_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p3, p4, p5, p6, p7, p8}
  {p2}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r10, r11, r12, r14, r15, r16, r17, r19, r20, r21, r22, r24, r26, r27, r28, r29, r30, r32, r33, r35, r36, r37, r38, r39, r40, r43, r47, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 3
          cons_2_A(x1,x2) = x1 + x2 + 3
          app#_A(x1,x2) = x1 + x2 + 4
          filter_1_A(x1) = x1 + 2
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5
          app_A(x1,x2) = x1 + x2 + 7
          true_0_A() = 1
          false_0_A() = 2
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          edge_0_A() = 1
          edge_1_A(x1) = x1 + 2
          edge_2_A(x1,x2) = x1 + x2 + 3
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 2
          eq_2_A(x1,x2) = x1 + x2 + 5
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          if_reach_1_0_A() = 1
          if_reach_1_1_A(x1) = x1 + 2
          if_reach_1_2_A(x1,x2) = x1 + x2 + 3
          if_reach_1_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          if_reach_2_0_A() = 1
          if_reach_2_1_A(x1) = x1 + 2
          if_reach_2_2_A(x1,x2) = x1 + x2 + 3
          if_reach_2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_2_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          or_0_A() = 1
          or_1_A(x1) = x1 + 2
          or_2_A(x1,x2) = x1 + x2 + 3
          reach_0_A() = 1
          reach_1_A(x1) = x1 + 2
          reach_2_A(x1,x2) = x1 + x2 + 3
          reach_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          union_0_A() = 1
          union_1_A(x1) = x1 + 2
          |0|_0_A() = 3
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 6
          empty_0_A() = 1
          if_reach_2_5_A(x1,x2,x3,x4,x5) = x4 + 1
          filter_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app# > filter2_4# > cons_2 = filter2_3 = filter_2 > app = edge_0 = nil_0 > filter_2# = cons_1 = filter2_2 = if_reach_1_4 = if_reach_2_4 = map_1 = reach_4 > filter_1 = cons_0 = eq_2 = filter2_1 = if_reach_1_3 = if_reach_2_3 = map_0 = or_2 = reach_3 > edge_3 = eq_1 = filter_0 = filter2_0 = if_reach_1_2 = if_reach_2_2 = or_1 = reach_2 = s_1 = |0|_0 > false_0 = edge_2 = eq_0 = if_reach_1_1 = if_reach_2_1 = or_0 = reach_1 = s_0 = union_1 = if_reach_2_5 > true_0 = edge_1 = if_reach_1_0 = if_reach_2_0 = reach_0 = union_0 = empty_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(app#) = [2]
      pi(filter_1) = [1]
      pi(filter2_4#) = []
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(false_0) = []
      pi(filter2_3) = [1]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(edge_0) = []
      pi(edge_1) = []
      pi(edge_2) = [1, 2]
      pi(edge_3) = [1, 3]
      pi(eq_0) = []
      pi(eq_1) = [1]
      pi(eq_2) = [1]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1]
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = [1]
      pi(if_reach_1_2) = [1, 2]
      pi(if_reach_1_3) = [3]
      pi(if_reach_1_4) = [1]
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = [1]
      pi(if_reach_2_2) = [1, 2]
      pi(if_reach_2_3) = [1, 2, 3]
      pi(if_reach_2_4) = [4]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(or_0) = []
      pi(or_1) = [1]
      pi(or_2) = []
      pi(reach_0) = []
      pi(reach_1) = [1]
      pi(reach_2) = [1, 2]
      pi(reach_3) = [3]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(union_0) = []
      pi(union_1) = [1]
      pi(|0|_0) = []
      pi(reach_4) = []
      pi(empty_0) = []
      pi(if_reach_2_5) = [4]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r10, r11, r12, r14, r15, r16, r17, r19, r20, r21, r22, r24, r25, r26, r27, r28, r29, r30, r32, r33, r35, r36, r37, r38, r39, r40, r43, r47, r49, r50, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x2 + 2
          cons_2_A(x1,x2) = x2 + 4
          filter2_4#_A(x1,x2,x3,x4) = x4 + 3
          app_A(x1,x2) = x1 + x2 + 10
          false_0_A() = 1
          true_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          edge_0_A() = 1
          edge_1_A(x1) = x1 + 2
          edge_2_A(x1,x2) = x1 + x2 + 4
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 3
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 2
          eq_2_A(x1,x2) = x1 + x2 + 6
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_0_A() = 1
          if_reach_1_1_A(x1) = x1 + 2
          if_reach_1_2_A(x1,x2) = x1 + x2 + 3
          if_reach_1_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          if_reach_2_0_A() = 1
          if_reach_2_1_A(x1) = x1 + 2
          if_reach_2_2_A(x1,x2) = x1 + x2 + 3
          if_reach_2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_2_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          map_2_A(x1,x2) = x1 + x2 + 5
          or_0_A() = 2
          or_1_A(x1) = x1 + 1
          or_2_A(x1,x2) = x1 + x2 + 2
          reach_0_A() = 1
          reach_1_A(x1) = x1 + 2
          reach_2_A(x1,x2) = x1 + x2 + 3
          reach_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          union_0_A() = 1
          union_1_A(x1) = x1 + 2
          |0|_0_A() = 2
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          empty_0_A() = 2
          if_reach_2_5_A(x1,x2,x3,x4,x5) = x4 + 1
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 3
  
    base order 2:
  
      precedence:
      
        app > or_2 = reach_4 = filter_2 > eq_2 = filter2_3 = map_2 = reach_3 = if_reach_2_5 > filter_2# = cons_2 = filter2_4# = edge_3 = eq_1 = filter_1 = filter2_2 = if_reach_2_4 = map_1 = or_1 = reach_2 = s_1 = union_1 = |0|_0 = nil_0 > true_0 = cons_1 = edge_2 = eq_0 = filter_0 = filter2_1 = if_reach_2_3 = map_0 = or_0 = reach_1 = s_0 = union_0 = empty_0 > false_0 = cons_0 = edge_1 = filter2_0 = if_reach_1_4 = if_reach_2_2 = reach_0 > edge_0 = if_reach_1_3 = if_reach_2_1 > if_reach_1_2 = if_reach_2_0 > if_reach_1_1 > if_reach_1_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(filter2_4#) = []
      pi(app) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(edge_0) = []
      pi(edge_1) = [1]
      pi(edge_2) = [2]
      pi(edge_3) = [1, 2, 3]
      pi(eq_0) = []
      pi(eq_1) = []
      pi(eq_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = [3]
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = [1]
      pi(if_reach_1_2) = []
      pi(if_reach_1_3) = []
      pi(if_reach_1_4) = [4]
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = [1]
      pi(if_reach_2_2) = []
      pi(if_reach_2_3) = []
      pi(if_reach_2_4) = [4]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(map_2) = []
      pi(or_0) = []
      pi(or_1) = [1]
      pi(or_2) = []
      pi(reach_0) = []
      pi(reach_1) = [1]
      pi(reach_2) = [2]
      pi(reach_3) = [2]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(union_0) = []
      pi(union_1) = [1]
      pi(|0|_0) = []
      pi(reach_4) = []
      pi(empty_0) = []
      pi(if_reach_2_5) = [4]
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,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(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,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(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r6, r8, r10, r11, r12, r14, r15, r17, r20, r22, r24, r26, r28, r29, r30, r33, r35, r36, r37, r38, r39, r40, r43, r47, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x1 + x2 + 6
          app_A(x1,x2) = 7
          cons_0_A() = 2
          cons_1_A(x1) = 1
          edge_0_A() = 3
          edge_1_A(x1) = 2
          edge_2_A(x1,x2) = 1
          eq_0_A() = 2
          eq_1_A(x1) = 1
          filter_0_A() = 7
          filter_1_A(x1) = 6
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          if_reach_1_0_A() = 3
          if_reach_1_1_A(x1) = 2
          if_reach_1_2_A(x1,x2) = 1
          if_reach_1_3_A(x1,x2,x3) = x1 + x3 + 8
          if_reach_1_4_A(x1,x2,x3,x4) = 1
          if_reach_2_1_A(x1) = x1 + 8
          if_reach_2_2_A(x1,x2) = 1
          if_reach_2_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
          if_reach_2_4_A(x1,x2,x3,x4) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          or_0_A() = 2
          or_1_A(x1) = 1
          reach_0_A() = 4
          reach_1_A(x1) = 3
          reach_2_A(x1,x2) = 2
          reach_3_A(x1,x2,x3) = 1
          union_0_A() = 2
          union_1_A(x1) = 1
          eq_2_A(x1,x2) = x1 + x2 + 12
          |0|_0_A() = 5
          true_0_A() = 4
          s_1_A(x1) = x1 + 8
          false_0_A() = 2
          or_2_A(x1,x2) = x1 + x2 + 8
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 13
          empty_0_A() = 1
          if_reach_2_5_A(x1,x2,x3,x4,x5) = x4 + 9
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 3
          filter_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        reach_4 > cons_2 = |0|_0 = or_2 = filter_2 > eq_2 = true_0 = edge_3 > map_2# = if_reach_1_3 = if_reach_2_3 = s_1 = if_reach_2_5 = nil_0 > app = cons_0 = edge_0 = eq_0 = filter_0 = filter2_0 = if_reach_1_0 = map_0 = or_0 = reach_0 = union_0 = false_0 > cons_1 = edge_1 = eq_1 = filter_1 = filter2_1 = if_reach_1_1 = if_reach_1_4 = if_reach_2_1 = if_reach_2_4 = map_1 = or_1 = reach_1 = union_1 = empty_0 > edge_2 = filter2_2 = if_reach_1_2 = if_reach_2_2 = reach_2 > filter2_3 = reach_3
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(edge_0) = []
      pi(edge_1) = []
      pi(edge_2) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = []
      pi(if_reach_1_2) = []
      pi(if_reach_1_3) = []
      pi(if_reach_1_4) = []
      pi(if_reach_2_1) = [1]
      pi(if_reach_2_2) = []
      pi(if_reach_2_3) = []
      pi(if_reach_2_4) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(reach_0) = []
      pi(reach_1) = []
      pi(reach_2) = []
      pi(reach_3) = []
      pi(union_0) = []
      pi(union_1) = []
      pi(eq_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_0) = []
      pi(or_2) = []
      pi(reach_4) = []
      pi(empty_0) = []
      pi(if_reach_2_5) = []
      pi(edge_3) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(v,y,union_2(i,h),empty_0())
p2: reach_4#(x,y,edge_3(u,v,i),h) -> if_reach_1_5#(eq_2(x,u),x,y,edge_3(u,v,i),h)
p3: if_reach_1_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,edge_3(u,v,h))
p4: if_reach_1_5#(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5#(eq_2(y,v),x,y,edge_3(u,v,i),h)
p5: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,h)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r6, r8, r10, r11, r14, r15, r16, r17, r19, r20, r21, r22, r24, r26, r28, r29, r30, r32, r33, r35, r36, r37, r38, r39, r40, r41, r42, r43, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_reach_2_5#_A(x1,x2,x3,x4,x5) = x4 + x5 + 13
          false_0_A() = 1
          edge_3_A(x1,x2,x3) = x1 + x3 + 13
          reach_4#_A(x1,x2,x3,x4) = x3 + x4 + 14
          union_2_A(x1,x2) = x1 + x2 + 9
          empty_0_A() = 2
          if_reach_1_5#_A(x1,x2,x3,x4,x5) = x4 + x5 + 14
          eq_2_A(x1,x2) = x1 + x2 + 28
          true_0_A() = 27
          app_A(x1,x2) = x2 + 8
          cons_0_A() = 2
          cons_1_A(x1) = 1
          edge_0_A() = 3
          edge_1_A(x1) = 2
          edge_2_A(x1,x2) = 1
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = 1
          if_reach_1_0_A() = 1
          if_reach_1_1_A(x1) = 4
          if_reach_1_2_A(x1,x2) = 3
          if_reach_1_3_A(x1,x2,x3) = 2
          if_reach_1_4_A(x1,x2,x3,x4) = 1
          if_reach_2_0_A() = 8
          if_reach_2_1_A(x1) = 7
          if_reach_2_2_A(x1,x2) = 6
          if_reach_2_3_A(x1,x2,x3) = 5
          if_reach_2_4_A(x1,x2,x3,x4) = 4
          map_0_A() = 2
          map_1_A(x1) = 1
          or_0_A() = 1
          or_1_A(x1) = 2
          reach_0_A() = 4
          reach_1_A(x1) = 3
          reach_2_A(x1,x2) = 2
          reach_3_A(x1,x2,x3) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          union_0_A() = 2
          union_1_A(x1) = 1
          |0|_0_A() = 28
          or_2_A(x1,x2) = x1 + x2 + 9
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + 23
          filter_2_A(x1,x2) = x1 + x2 + 28
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        if_reach_2_5# = filter_2 > reach_4# = if_reach_1_5# = eq_2 = nil_0 > union_2 = if_reach_1_2 = or_2 = reach_4 > app = cons_0 = edge_0 = filter_0 = if_reach_2_0 = map_0 = reach_0 = union_0 > cons_1 = edge_1 = eq_1 = filter_1 = filter2_1 = if_reach_1_1 = if_reach_1_3 = if_reach_2_1 = map_1 = or_1 = reach_1 = union_1 > eq_0 = filter2_0 = filter2_2 = if_reach_1_0 = if_reach_1_4 = if_reach_2_2 = or_0 = reach_2 > if_reach_2_3 = reach_3 = |0|_0 > true_0 = if_reach_2_4 > empty_0 = s_1 > false_0 = s_0 > edge_3 > edge_2
      
  
    partial status:
  
      pi(if_reach_2_5#) = [4, 5]
      pi(false_0) = []
      pi(edge_3) = []
      pi(reach_4#) = [3]
      pi(union_2) = [1, 2]
      pi(empty_0) = []
      pi(if_reach_1_5#) = [4]
      pi(eq_2) = []
      pi(true_0) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(edge_0) = []
      pi(edge_1) = []
      pi(edge_2) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = []
      pi(if_reach_1_2) = []
      pi(if_reach_1_3) = []
      pi(if_reach_1_4) = []
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = []
      pi(if_reach_2_2) = []
      pi(if_reach_2_3) = []
      pi(if_reach_2_4) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(reach_0) = []
      pi(reach_1) = []
      pi(reach_2) = []
      pi(reach_3) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(union_0) = []
      pi(union_1) = []
      pi(|0|_0) = []
      pi(or_2) = [1, 2]
      pi(reach_4) = [1, 3]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(v,y,union_2(i,h),empty_0())
p2: reach_4#(x,y,edge_3(u,v,i),h) -> if_reach_1_5#(eq_2(x,u),x,y,edge_3(u,v,i),h)
p3: if_reach_1_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,edge_3(u,v,h))
p4: if_reach_1_5#(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5#(eq_2(y,v),x,y,edge_3(u,v,i),h)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: if_reach_2_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(v,y,union_2(i,h),empty_0())
p2: reach_4#(x,y,edge_3(u,v,i),h) -> if_reach_1_5#(eq_2(x,u),x,y,edge_3(u,v,i),h)
p3: if_reach_1_5#(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5#(eq_2(y,v),x,y,edge_3(u,v,i),h)
p4: if_reach_1_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,edge_3(u,v,h))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r8, r10, r11, r12, r14, r16, r17, r19, r20, r21, r22, r24, r26, r27, r28, r29, r30, r32, r33, r34, r35, r36, r37, r39, r40, r41, r42, r43, r47, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_reach_2_5#_A(x1,x2,x3,x4,x5) = x4 + x5 + 2
          false_0_A() = 1
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 8
          reach_4#_A(x1,x2,x3,x4) = x1 + x3 + x4 + 3
          union_2_A(x1,x2) = x1 + x2 + 4
          empty_0_A() = 2
          if_reach_1_5#_A(x1,x2,x3,x4,x5) = x2 + x4 + x5 + 3
          eq_2_A(x1,x2) = x1 + x2 + 3
          true_0_A() = 1
          app_A(x1,x2) = x1 + x2 + 6
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 3
          cons_2_A(x1,x2) = x1 + 2
          edge_0_A() = 2
          edge_1_A(x1) = x1 + 1
          edge_2_A(x1,x2) = x1 + x2 + 3
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_0_A() = 1
          if_reach_1_1_A(x1) = x1 + 2
          if_reach_1_2_A(x1,x2) = x1 + x2 + 9
          if_reach_1_3_A(x1,x2,x3) = x1 + x2 + x3 + 10
          if_reach_1_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 11
          if_reach_2_0_A() = 1
          if_reach_2_1_A(x1) = x1 + 2
          if_reach_2_2_A(x1,x2) = x1 + x2 + 3
          if_reach_2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_2_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          or_0_A() = 1
          or_1_A(x1) = x1 + 2
          or_2_A(x1,x2) = x1 + x2 + 3
          reach_0_A() = 1
          reach_1_A(x1) = x1 + 2
          reach_2_A(x1,x2) = x1 + x2 + 3
          reach_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          union_0_A() = 1
          union_1_A(x1) = x1 + 2
          |0|_0_A() = 2
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 9
          if_reach_2_5_A(x1,x2,x3,x4,x5) = x1 + 2
          filter_2_A(x1,x2) = x2 + 3
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        reach_4# = if_reach_1_5# = eq_2 > if_reach_2_5# = reach_4 > union_2 = or_2 = reach_3 > reach_2 = union_1 = if_reach_2_5 > app = eq_0 = if_reach_1_0 = union_0 > cons_1 = eq_1 = filter2_3 = if_reach_2_4 = or_1 = reach_1 = s_1 > edge_3 = cons_0 = cons_2 = filter2_2 = if_reach_1_4 = if_reach_2_3 = or_0 = reach_0 = s_0 = |0|_0 = filter_2 > empty_0 = true_0 = edge_2 = filter_1 = filter2_1 = if_reach_1_3 = if_reach_2_2 = map_1 = nil_0 > false_0 = edge_1 = filter_0 = filter2_0 = if_reach_1_2 = if_reach_2_1 = map_0 > edge_0 = if_reach_1_1 = if_reach_2_0
      
  
    partial status:
  
      pi(if_reach_2_5#) = []
      pi(false_0) = []
      pi(edge_3) = [2]
      pi(reach_4#) = [3]
      pi(union_2) = []
      pi(empty_0) = []
      pi(if_reach_1_5#) = [4]
      pi(eq_2) = []
      pi(true_0) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(cons_2) = []
      pi(edge_0) = []
      pi(edge_1) = [1]
      pi(edge_2) = [2]
      pi(eq_0) = []
      pi(eq_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(filter2_3) = [1]
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = []
      pi(if_reach_1_2) = [1]
      pi(if_reach_1_3) = [1]
      pi(if_reach_1_4) = [4]
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = [1]
      pi(if_reach_2_2) = [1, 2]
      pi(if_reach_2_3) = [1, 2, 3]
      pi(if_reach_2_4) = [4]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(or_0) = []
      pi(or_1) = [1]
      pi(or_2) = []
      pi(reach_0) = []
      pi(reach_1) = [1]
      pi(reach_2) = [1, 2]
      pi(reach_3) = [3]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(union_0) = []
      pi(union_1) = [1]
      pi(|0|_0) = []
      pi(reach_4) = []
      pi(if_reach_2_5) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: reach_4#(x,y,edge_3(u,v,i),h) -> if_reach_1_5#(eq_2(x,u),x,y,edge_3(u,v,i),h)
p2: if_reach_1_5#(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5#(eq_2(y,v),x,y,edge_3(u,v,i),h)
p3: if_reach_1_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,edge_3(u,v,h))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p3}


-- Reduction pair.

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

p1: reach_4#(x,y,edge_3(u,v,i),h) -> if_reach_1_5#(eq_2(x,u),x,y,edge_3(u,v,i),h)
p2: if_reach_1_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,edge_3(u,v,h))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r6, r7, r8, r10, r11, r12, r14, r15, r16, r19, r20, r21, r22, r24, r26, r28, r29, r30, r32, r33, r35, r36, r37, r38, r39, r40, r43, r47, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          reach_4#_A(x1,x2,x3,x4) = x3 + 3
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_5#_A(x1,x2,x3,x4,x5) = x4 + 1
          eq_2_A(x1,x2) = x2 + 2
          false_0_A() = 2
          app_A(x1,x2) = x2 + 5
          cons_0_A() = 2
          cons_1_A(x1) = 1
          edge_0_A() = 3
          edge_1_A(x1) = 2
          edge_2_A(x1,x2) = 1
          eq_0_A() = 2
          eq_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          if_reach_1_0_A() = 1
          if_reach_1_1_A(x1) = x1 + 2
          if_reach_1_2_A(x1,x2) = 1
          if_reach_1_3_A(x1,x2,x3) = x3 + 2
          if_reach_2_0_A() = 5
          if_reach_2_1_A(x1) = 4
          if_reach_2_2_A(x1,x2) = 3
          if_reach_2_3_A(x1,x2,x3) = 2
          if_reach_2_4_A(x1,x2,x3,x4) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          or_0_A() = 2
          or_1_A(x1) = 1
          reach_0_A() = 4
          reach_1_A(x1) = 3
          reach_2_A(x1,x2) = 2
          reach_3_A(x1,x2,x3) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          union_0_A() = 2
          union_1_A(x1) = 1
          |0|_0_A() = 3
          true_0_A() = 1
          or_2_A(x1,x2) = x1 + x2 + 6
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 11
          empty_0_A() = 1
          if_reach_2_5_A(x1,x2,x3,x4,x5) = x4 + 6
          filter_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        if_reach_2_0 > if_reach_2_1 > if_reach_2_2 > if_reach_2_3 > if_reach_2_4 = or_2 > union_0 = if_reach_2_5 > union_1 = filter_2 > app = filter2_0 = reach_0 = nil_0 > if_reach_1_5# = eq_2 = cons_0 = eq_0 = filter_0 = filter2_1 = if_reach_1_1 = map_0 = or_0 = reach_1 = s_1 > edge_3 = cons_1 = eq_1 = filter_1 = filter2_2 = if_reach_1_0 = if_reach_1_2 = map_1 = or_1 = reach_2 = s_0 = |0|_0 = reach_4 > reach_4# = edge_0 = edge_2 = filter2_3 = if_reach_1_3 = reach_3 = true_0 > false_0 = edge_1 > empty_0
      
  
    partial status:
  
      pi(reach_4#) = []
      pi(edge_3) = []
      pi(if_reach_1_5#) = [4]
      pi(eq_2) = []
      pi(false_0) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(edge_0) = []
      pi(edge_1) = []
      pi(edge_2) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = []
      pi(if_reach_1_2) = []
      pi(if_reach_1_3) = [3]
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = []
      pi(if_reach_2_2) = []
      pi(if_reach_2_3) = []
      pi(if_reach_2_4) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(reach_0) = []
      pi(reach_1) = []
      pi(reach_2) = []
      pi(reach_3) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(union_0) = []
      pi(union_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(or_2) = [2]
      pi(reach_4) = [1, 3]
      pi(empty_0) = []
      pi(if_reach_2_5) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: if_reach_1_5#(false_0(),x,y,edge_3(u,v,i),h) -> reach_4#(x,y,i,edge_3(u,v,h))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: eq_2#(s_1(x),s_1(y)) -> eq_2#(x,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r10, r11, r12, r14, r15, r16, r17, r19, r20, r21, r22, r24, r26, r27, r28, r29, r30, r32, r33, r35, r36, r37, r38, r39, r40, r43, r47, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          eq_2#_A(x1,x2) = x1 + x2 + 4
          s_1_A(x1) = x1 + 3
          app_A(x1,x2) = x1 + x2 + 6
          cons_0_A() = 2
          cons_1_A(x1) = x1 + 1
          cons_2_A(x1,x2) = x1 + x2 + 2
          edge_0_A() = 1
          edge_1_A(x1) = x1 + 2
          edge_2_A(x1,x2) = x1 + x2 + 3
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          eq_0_A() = 1
          eq_1_A(x1) = x1 + 2
          eq_2_A(x1,x2) = x1 + x2 + 5
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 4
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_0_A() = 1
          if_reach_1_1_A(x1) = x1 + 2
          if_reach_1_2_A(x1,x2) = x1 + x2 + 3
          if_reach_1_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          if_reach_1_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 5
          if_reach_2_0_A() = 1
          if_reach_2_1_A(x1) = x1 + 2
          if_reach_2_2_A(x1,x2) = x1 + x2 + 1
          if_reach_2_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
          if_reach_2_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 3
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          or_0_A() = 1
          or_1_A(x1) = x1 + 2
          or_2_A(x1,x2) = x1 + x2 + 3
          reach_0_A() = 1
          reach_1_A(x1) = x1 + 2
          reach_2_A(x1,x2) = x1 + x2 + 3
          reach_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          s_0_A() = 1
          union_0_A() = 1
          union_1_A(x1) = x1 + 2
          |0|_0_A() = 3
          true_0_A() = 1
          false_0_A() = 2
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 6
          empty_0_A() = 1
          if_reach_2_5_A(x1,x2,x3,x4,x5) = x4 + 1
          filter_2_A(x1,x2) = x1 + x2 + 3
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        edge_2 > eq_2# = app = cons_2 > s_1 = filter2_3 = or_2 = reach_4 = filter_2 > false_0 = nil_0 > eq_1 = eq_2 = |0|_0 = if_reach_2_5 > filter_0 = if_reach_1_2 = if_reach_1_4 = if_reach_2_4 = true_0 > cons_1 = edge_1 = edge_3 = eq_0 = filter_1 = filter2_2 = if_reach_1_1 = if_reach_1_3 = if_reach_2_3 = map_1 = or_1 = reach_3 = s_0 > cons_0 = edge_0 = filter2_1 = if_reach_1_0 = if_reach_2_2 = map_0 = or_0 = reach_2 = union_1 = empty_0 > filter2_0 = if_reach_2_1 = reach_1 = union_0 > if_reach_2_0 = reach_0
      
  
    partial status:
  
      pi(eq_2#) = [1]
      pi(s_1) = [1]
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(cons_2) = [2]
      pi(edge_0) = []
      pi(edge_1) = [1]
      pi(edge_2) = [2]
      pi(edge_3) = [3]
      pi(eq_0) = []
      pi(eq_1) = [1]
      pi(eq_2) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(filter2_3) = [3]
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = [1]
      pi(if_reach_1_2) = [2]
      pi(if_reach_1_3) = []
      pi(if_reach_1_4) = []
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = [1]
      pi(if_reach_2_2) = [2]
      pi(if_reach_2_3) = []
      pi(if_reach_2_4) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(or_0) = []
      pi(or_1) = [1]
      pi(or_2) = []
      pi(reach_0) = []
      pi(reach_1) = [1]
      pi(reach_2) = [2]
      pi(reach_3) = [3]
      pi(s_0) = []
      pi(union_0) = []
      pi(union_1) = [1]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(reach_4) = [3]
      pi(empty_0) = []
      pi(if_reach_2_5) = [4]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: union_2#(edge_3(x,y,i),h) -> union_2#(i,h)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(edge_0(),y) -> edge_1(y)
r4: app(edge_1(x1),y) -> edge_2(x1,y)
r5: app(edge_2(x1,x2),y) -> edge_3(x1,x2,y)
r6: app(eq_0(),y) -> eq_1(y)
r7: app(eq_1(x1),y) -> eq_2(x1,y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(if_reach_1_0(),y) -> if_reach_1_1(y)
r15: app(if_reach_1_1(x1),y) -> if_reach_1_2(x1,y)
r16: app(if_reach_1_2(x1,x2),y) -> if_reach_1_3(x1,x2,y)
r17: app(if_reach_1_3(x1,x2,x3),y) -> if_reach_1_4(x1,x2,x3,y)
r18: app(if_reach_1_4(x1,x2,x3,x4),y) -> if_reach_1_5(x1,x2,x3,x4,y)
r19: app(if_reach_2_0(),y) -> if_reach_2_1(y)
r20: app(if_reach_2_1(x1),y) -> if_reach_2_2(x1,y)
r21: app(if_reach_2_2(x1,x2),y) -> if_reach_2_3(x1,x2,y)
r22: app(if_reach_2_3(x1,x2,x3),y) -> if_reach_2_4(x1,x2,x3,y)
r23: app(if_reach_2_4(x1,x2,x3,x4),y) -> if_reach_2_5(x1,x2,x3,x4,y)
r24: app(map_0(),y) -> map_1(y)
r25: app(map_1(x1),y) -> map_2(x1,y)
r26: app(or_0(),y) -> or_1(y)
r27: app(or_1(x1),y) -> or_2(x1,y)
r28: app(reach_0(),y) -> reach_1(y)
r29: app(reach_1(x1),y) -> reach_2(x1,y)
r30: app(reach_2(x1,x2),y) -> reach_3(x1,x2,y)
r31: app(reach_3(x1,x2,x3),y) -> reach_4(x1,x2,x3,y)
r32: app(s_0(),y) -> s_1(y)
r33: app(union_0(),y) -> union_1(y)
r34: app(union_1(x1),y) -> union_2(x1,y)
r35: eq_2(|0|_0(),|0|_0()) -> true_0()
r36: eq_2(|0|_0(),s_1(x)) -> false_0()
r37: eq_2(s_1(x),|0|_0()) -> false_0()
r38: eq_2(s_1(x),s_1(y)) -> eq_2(x,y)
r39: or_2(true_0(),y) -> true_0()
r40: or_2(false_0(),y) -> y
r41: union_2(empty_0(),h) -> h
r42: union_2(edge_3(x,y,i),h) -> edge_3(x,y,union_2(i,h))
r43: reach_4(x,y,empty_0(),h) -> false_0()
r44: reach_4(x,y,edge_3(u,v,i),h) -> if_reach_1_5(eq_2(x,u),x,y,edge_3(u,v,i),h)
r45: if_reach_1_5(true_0(),x,y,edge_3(u,v,i),h) -> if_reach_2_5(eq_2(y,v),x,y,edge_3(u,v,i),h)
r46: if_reach_1_5(false_0(),x,y,edge_3(u,v,i),h) -> reach_4(x,y,i,edge_3(u,v,h))
r47: if_reach_2_5(true_0(),x,y,edge_3(u,v,i),h) -> true_0()
r48: if_reach_2_5(false_0(),x,y,edge_3(u,v,i),h) -> or_2(reach_4(x,y,i,h),reach_4(v,y,union_2(i,h),empty_0()))
r49: map_2(f,nil_0()) -> nil_0()
r50: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r51: filter_2(f,nil_0()) -> nil_0()
r52: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r53: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r54: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r6, r8, r10, r11, r12, r14, r15, r19, r20, r21, r22, r24, r26, r28, r29, r30, r32, r33, r35, r36, r37, r38, r39, r40, r43, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          union_2#_A(x1,x2) = x1 + x2 + 9
          edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 8
          app_A(x1,x2) = x2 + 7
          cons_0_A() = 2
          cons_1_A(x1) = 1
          edge_0_A() = 3
          edge_1_A(x1) = 2
          edge_2_A(x1,x2) = 1
          eq_0_A() = 2
          eq_1_A(x1) = 1
          filter_0_A() = 6
          filter_1_A(x1) = 5
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          if_reach_1_0_A() = 3
          if_reach_1_1_A(x1) = 2
          if_reach_1_2_A(x1,x2) = 1
          if_reach_2_0_A() = 7
          if_reach_2_1_A(x1) = x1 + 6
          if_reach_2_2_A(x1,x2) = 5
          if_reach_2_3_A(x1,x2,x3) = 4
          if_reach_2_4_A(x1,x2,x3,x4) = 3
          map_0_A() = 2
          map_1_A(x1) = 1
          or_0_A() = 2
          or_1_A(x1) = 1
          reach_0_A() = 4
          reach_1_A(x1) = 3
          reach_2_A(x1,x2) = 2
          reach_3_A(x1,x2,x3) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          union_0_A() = 2
          union_1_A(x1) = 1
          eq_2_A(x1,x2) = x1 + x2 + 19
          |0|_0_A() = 4
          true_0_A() = 3
          false_0_A() = 2
          or_2_A(x1,x2) = x1 + x2 + 8
          reach_4_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 12
          empty_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        reach_0 = |0|_0 = or_2 = filter_2 > union_2# = reach_1 = eq_2 = true_0 = reach_4 > edge_3 = reach_2 > reach_3 > app = cons_0 = edge_0 = eq_0 = filter_0 = filter2_0 = if_reach_1_0 = map_0 = or_0 = union_0 = nil_0 > cons_1 = edge_1 = eq_1 = filter_1 = filter2_1 = if_reach_1_1 = if_reach_2_2 = map_1 = or_1 = s_1 = union_1 > edge_2 = filter2_2 = if_reach_1_2 = if_reach_2_1 = if_reach_2_3 = s_0 = false_0 > filter2_3 = if_reach_2_0 = if_reach_2_4 > empty_0
      
  
    partial status:
  
      pi(union_2#) = [1, 2]
      pi(edge_3) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(edge_0) = []
      pi(edge_1) = []
      pi(edge_2) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = []
      pi(if_reach_1_2) = []
      pi(if_reach_2_0) = []
      pi(if_reach_2_1) = [1]
      pi(if_reach_2_2) = []
      pi(if_reach_2_3) = []
      pi(if_reach_2_4) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(reach_0) = []
      pi(reach_1) = []
      pi(reach_2) = []
      pi(reach_3) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(union_0) = []
      pi(union_1) = []
      pi(eq_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(or_2) = []
      pi(reach_4) = [4]
      pi(empty_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.