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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x1 + x2
              cons_2_A(x1,x2) = x1 + x2 + 7
              app#_A(x1,x2) = x1 + x2
              map_1_A(x1) = x1 + 36
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 39
              filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 7
              false_0_A() = 70
              filter_2#_A(x1,x2) = x1 + x2 + 7
              filter_1_A(x1) = x1 + 7
              app_A(x1,x2) = x1 + x2 + 5
              true_0_A() = 41
              map_2#!_A(x1,x2) = x2 + 30
              cons_2!_A(x1,x2) = x1 + x2 + 37
              app#!_A(x1,x2) = x2 + 36
              map_1!_A(x1) = 31
              filter2_3!_A(x1,x2,x3) = x1 + x2 + 34
              filter2_4#!_A(x1,x2,x3,x4) = x4 + 35
              false_0!_A() = 24
              filter_2#!_A(x1,x2) = x2 + 32
              filter_1!_A(x1) = 31
              app!_A(x1,x2) = x1 + 2
              true_0!_A() = 29
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x1 + x2
              cons_2_A(x1,x2) = x1 + x2 + 10
              app#_A(x1,x2) = x1 + x2
              map_1_A(x1) = x1 + 3
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
              filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 3
              false_0_A() = 15
              filter_2#_A(x1,x2) = x1 + x2 + 3
              filter_1_A(x1) = x1 + 3
              app_A(x1,x2) = x1 + 1
              true_0_A() = 13
              map_2#!_A(x1,x2) = x2 + 19
              cons_2!_A(x1,x2) = x1 + x2 + 27
              app#!_A(x1,x2) = x2 + 20
              map_1!_A(x1) = 2
              filter2_3!_A(x1,x2,x3) = 2
              filter2_4#!_A(x1,x2,x3,x4) = 18
              false_0!_A() = 1
              filter_2#!_A(x1,x2) = x2 + 19
              filter_1!_A(x1) = 1
              app!_A(x1,x2) = 26
              true_0!_A() = 18
        
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app#) = [2]
      pi(map_1) = [1]
      pi(filter2_3) = [3]
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = [1]
      pi(app) = [1, 2]
      pi(true_0) = []

The following rules are removed:

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

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r35, r36, r37, r38, r41, r42

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              if_reach_2_5#_A(x1,x2,x3,x4,x5) = 0
              false_0_A() = 4
              edge_3_A(x1,x2,x3) = x2 + x3 + 6
              reach_4#_A(x1,x2,x3,x4) = 0
              union_2_A(x1,x2) = x1 + x2 + 2
              empty_0_A() = 1
              if_reach_1_5#_A(x1,x2,x3,x4,x5) = 0
              eq_2_A(x1,x2) = 5
              true_0_A() = 5
              |0|_0_A() = 1
              s_1_A(x1) = 2
              if_reach_2_5#!_A(x1,x2,x3,x4,x5) = x2 + x4 + x5 + 2
              false_0!_A() = 4
              edge_3!_A(x1,x2,x3) = x2 + 9
              reach_4#!_A(x1,x2,x3,x4) = x1 + x3 + x4 + 3
              union_2!_A(x1,x2) = x1 + x2 + 14
              empty_0!_A() = 0
              if_reach_1_5#!_A(x1,x2,x3,x4,x5) = x2 + x4 + x5 + 3
              eq_2!_A(x1,x2) = 8
              true_0!_A() = 4
              |0|_0!_A() = 5
              s_1!_A(x1) = 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          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
              false_0_A() = 7
              edge_3_A(x1,x2,x3) = x2 + x3 + 12
              reach_4#_A(x1,x2,x3,x4) = x3 + x4
              union_2_A(x1,x2) = x1 + x2 + 2
              empty_0_A() = 10
              if_reach_1_5#_A(x1,x2,x3,x4,x5) = x4 + x5
              eq_2_A(x1,x2) = x1 + x2 + 2
              true_0_A() = 5
              |0|_0_A() = 3
              s_1_A(x1) = x1 + 2
              if_reach_2_5#!_A(x1,x2,x3,x4,x5) = x1 + x2 + x4 + x5
              false_0!_A() = 17
              edge_3!_A(x1,x2,x3) = x2 + 26
              reach_4#!_A(x1,x2,x3,x4) = x1 + x3 + 14
              union_2!_A(x1,x2) = x1 + 27
              empty_0!_A() = 18
              if_reach_1_5#!_A(x1,x2,x3,x4,x5) = x2 + x4 + 3
              eq_2!_A(x1,x2) = 16
              true_0!_A() = 17
              |0|_0!_A() = 18
              s_1!_A(x1) = 15
        
  
    partial status:
  
      pi(if_reach_2_5#) = [1]
      pi(false_0) = []
      pi(edge_3) = []
      pi(reach_4#) = []
      pi(union_2) = [2]
      pi(empty_0) = []
      pi(if_reach_1_5#) = [1]
      pi(eq_2) = []
      pi(true_0) = []
      pi(|0|_0) = []
      pi(s_1) = []

The following rules are removed:

  p1, p2, p3, p4, p5

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: app(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

  r11, r14, r42

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              eq_2#_A(x1,x2) = x1 + x2
              s_1_A(x1) = x1 + 2
              app_A(x1,x2) = x2 + 1
              filter2_1_A(x1) = x1 + 2
              filter2_2_A(x1,x2) = 1
              if_reach_1_0_A() = 2
              if_reach_1_1_A(x1) = 1
              union_2_A(x1,x2) = x1 + 4
              edge_3_A(x1,x2,x3) = x1 + x2 + 2
              eq_2#!_A(x1,x2) = x1
              s_1!_A(x1) = 18
              app!_A(x1,x2) = x1 + x2
              filter2_1!_A(x1) = 2
              filter2_2!_A(x1,x2) = x1
              if_reach_1_0!_A() = 0
              if_reach_1_1!_A(x1) = 1
              union_2!_A(x1,x2) = x1 + 18
              edge_3!_A(x1,x2,x3) = x1 + x2 + 12
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              eq_2#_A(x1,x2) = x1 + x2
              s_1_A(x1) = x1 + 3
              app_A(x1,x2) = x1 + x2 + 1
              filter2_1_A(x1) = x1 + 3
              filter2_2_A(x1,x2) = x1 + x2 + 1
              if_reach_1_0_A() = 1
              if_reach_1_1_A(x1) = x1 + 1
              union_2_A(x1,x2) = x1 + 5
              edge_3_A(x1,x2,x3) = x1 + x2 + 7
              eq_2#!_A(x1,x2) = x1
              s_1!_A(x1) = 13
              app!_A(x1,x2) = 15
              filter2_1!_A(x1) = 0
              filter2_2!_A(x1,x2) = x2 + 1
              if_reach_1_0!_A() = 17
              if_reach_1_1!_A(x1) = x1 + 16
              union_2!_A(x1,x2) = x1 + 4
              edge_3!_A(x1,x2,x3) = x2 + 11
        
  
    partial status:
  
      pi(eq_2#) = [1]
      pi(s_1) = [1]
      pi(app) = [1, 2]
      pi(filter2_1) = []
      pi(filter2_2) = [2]
      pi(if_reach_1_0) = []
      pi(if_reach_1_1) = []
      pi(union_2) = [1, 2]
      pi(edge_3) = []

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

  r20, r24, r30, r51

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              union_2#_A(x1,x2) = x1
              edge_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
              app_A(x1,x2) = x2 + 1
              if_reach_2_1_A(x1) = 2
              if_reach_2_2_A(x1,x2) = 1
              map_0_A() = 1
              map_1_A(x1) = 1
              reach_2_A(x1,x2) = 2
              reach_3_A(x1,x2,x3) = 1
              filter_2_A(x1,x2) = 3
              nil_0_A() = 3
              union_2#!_A(x1,x2) = x1 + 8
              edge_3!_A(x1,x2,x3) = x1 + x2 + x3 + 7
              app!_A(x1,x2) = x2 + 5
              if_reach_2_1!_A(x1) = 4
              if_reach_2_2!_A(x1,x2) = 3
              map_0!_A() = 0
              map_1!_A(x1) = 1
              reach_2!_A(x1,x2) = 4
              reach_3!_A(x1,x2,x3) = x3 + 3
              filter_2!_A(x1,x2) = x1 + 3
              nil_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:
              union_2#_A(x1,x2) = x1
              edge_3_A(x1,x2,x3) = x2 + x3 + 16
              app_A(x1,x2) = x1 + x2 + 1
              if_reach_2_1_A(x1) = x1 + 3
              if_reach_2_2_A(x1,x2) = x1 + x2 + 1
              map_0_A() = 1
              map_1_A(x1) = x1 + 1
              reach_2_A(x1,x2) = x1 + x2 + 5
              reach_3_A(x1,x2,x3) = x2 + x3 + 1
              filter_2_A(x1,x2) = x2 + 12
              nil_0_A() = 4
              union_2#!_A(x1,x2) = x1
              edge_3!_A(x1,x2,x3) = x1 + x2 + x3 + 30
              app!_A(x1,x2) = x1 + x2 + 21
              if_reach_2_1!_A(x1) = 21
              if_reach_2_2!_A(x1,x2) = 20
              map_0!_A() = 0
              map_1!_A(x1) = 9
              reach_2!_A(x1,x2) = 25
              reach_3!_A(x1,x2,x3) = 24
              filter_2!_A(x1,x2) = x2 + 30
              nil_0!_A() = 0
        
  
    partial status:
  
      pi(union_2#) = [2]
      pi(edge_3) = []
      pi(app) = [2]
      pi(if_reach_2_1) = []
      pi(if_reach_2_2) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(reach_2) = [1, 2]
      pi(reach_3) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.