We consider the system Applicative_first_order_05__#3.13.

  Alphabet:

    0 : [] --> a 
    cons : [d * e] --> e 
    edge : [a * a * b] --> b 
    empty : [] --> b 
    eq : [a * a] --> c 
    false : [] --> c 
    filter : [d -> c * e] --> e 
    filter2 : [c * d -> c * d * e] --> e 
    if!fac6220reach!fac62201 : [c * a * a * b * b] --> c 
    if!fac6220reach!fac62202 : [c * a * a * b * b] --> c 
    map : [d -> d * e] --> e 
    nil : [] --> e 
    or : [c * c] --> c 
    reach : [a * a * b * b] --> c 
    s : [a] --> a 
    true : [] --> c 
    union : [b * b] --> b 

  Rules:

    eq(0, 0) => true 
    eq(0, s(x)) => false 
    eq(s(x), 0) => false 
    eq(s(x), s(y)) => eq(x, y) 
    or(true, x) => true 
    or(false, x) => x 
    union(empty, x) => x 
    union(edge(x, y, z), u) => edge(x, y, union(z, u)) 
    reach(x, y, empty, z) => false 
    reach(x, y, edge(z, u, v), w) => if!fac6220reach!fac62201(eq(x, z), x, y, edge(z, u, v), w) 
    if!fac6220reach!fac62201(true, x, y, edge(z, u, v), w) => if!fac6220reach!fac62202(eq(y, u), x, y, edge(z, u, v), w) 
    if!fac6220reach!fac62201(false, x, y, edge(z, u, v), w) => reach(x, y, v, edge(z, u, w)) 
    if!fac6220reach!fac62202(true, x, y, edge(z, u, v), w) => true 
    if!fac6220reach!fac62202(false, x, y, edge(z, u, v), w) => or(reach(x, y, v, w), reach(u, y, union(v, w), empty)) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
    filter2(true, f, x, y) => cons(x, filter(f, y)) 
    filter2(false, f, x, y) => filter(f, y) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We observe that the rules contain a first-order subset:

  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  or(true, X) => true 
  or(false, X) => X 
  union(empty, X) => X 
  union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) => false 
  reach(X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 

Moreover, the system is orthogonal.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is terminating when seen as a many-sorted first-order TRS.

According to mutermprover, this system is indeed terminating:

 || 
 || Problem 1: 
 || 
 || (VAR %U %V %W %X %Y %Z)
 || (RULES
 || eq(0,0) -> true
 || eq(0,s(%X)) -> false
 || eq(s(%X),0) -> false
 || eq(s(%X),s(%Y)) -> eq(%X,%Y)
 || if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 || if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 || if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 || if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 || or(false,%X) -> %X
 || or(true,%X) -> true
 || reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 || reach(%X,%Y,empty,%Z) -> false
 || union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 || union(empty,%X) -> %X
 || )
 || 
 || Problem 1: 
 || 
 || Innermost Equivalent Processor:
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 ||  
 || 
 || Problem 1: 
 || 
 || Dependency Pairs Processor:
 || -> Pairs:
 ||  EQ(s(%X),s(%Y)) -> EQ(%X,%Y)
 ||  IF!FAC6220REACH!FAC62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,edge(%Z,%U,%W))
 ||  IF!FAC6220REACH!FAC62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> EQ(%Y,%U)
 ||  IF!FAC6220REACH!FAC62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> OR(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%U,%Y,union(%V,%W),empty)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,%W)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> UNION(%V,%W)
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> EQ(%X,%Z)
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  UNION(edge(%X,%Y,%Z),%U) -> UNION(%Z,%U)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || 
 || Problem 1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  EQ(s(%X),s(%Y)) -> EQ(%X,%Y)
 ||  IF!FAC6220REACH!FAC62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,edge(%Z,%U,%W))
 ||  IF!FAC6220REACH!FAC62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> EQ(%Y,%U)
 ||  IF!FAC6220REACH!FAC62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> OR(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%U,%Y,union(%V,%W),empty)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,%W)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> UNION(%V,%W)
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> EQ(%X,%Z)
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  UNION(edge(%X,%Y,%Z),%U) -> UNION(%Z,%U)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Strongly Connected Components:
 || ->->Cycle:
 || ->->-> Pairs:
 ||  UNION(edge(%X,%Y,%Z),%U) -> UNION(%Z,%U)
 || ->->-> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->->Cycle:
 || ->->-> Pairs:
 ||  EQ(s(%X),s(%Y)) -> EQ(%X,%Y)
 || ->->-> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->->Cycle:
 || ->->-> Pairs:
 ||  IF!FAC6220REACH!FAC62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,edge(%Z,%U,%W))
 ||  IF!FAC6220REACH!FAC62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%U,%Y,union(%V,%W),empty)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,%W)
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 || ->->-> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || 
 || 
 || The problem is decomposed in 3 subproblems.
 || 
 || Problem 1.1: 
 || 
 || Subterm Processor:
 || -> Pairs:
 ||  UNION(edge(%X,%Y,%Z),%U) -> UNION(%Z,%U)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Projection:
 ||  pi(UNION) = 1
 || 
 || Problem 1.1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Strongly Connected Components:
 ||  There is no strongly connected component
 || 
 || The problem is finite.
 || 
 || Problem 1.2: 
 || 
 || Subterm Processor:
 || -> Pairs:
 ||  EQ(s(%X),s(%Y)) -> EQ(%X,%Y)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Projection:
 ||  pi(EQ) = 1
 || 
 || Problem 1.2: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Strongly Connected Components:
 ||  There is no strongly connected component
 || 
 || The problem is finite.
 || 
 || Problem 1.3: 
 || 
 || Reduction Pairs Processor:
 || -> Pairs:
 ||  IF!FAC6220REACH!FAC62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,edge(%Z,%U,%W))
 ||  IF!FAC6220REACH!FAC62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%U,%Y,union(%V,%W),empty)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,%W)
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || -> Usable rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Interpretation type:
 ||  Linear
 || ->Coefficients:
 ||  Natural Numbers
 || ->Dimension:
 ||  1
 || ->Bound:
 ||  2
 || ->Interpretation:
 ||  
 || [eq](X1,X2) = 2
 || [union](X1,X2) = X1 + X2 + 1
 || [0] = 0
 || [edge](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
 || [empty] = 0
 || [false] = 2
 || [s](X) = 2.X
 || [true] = 2
 || [IF!FAC6220REACH!FAC62201](X1,X2,X3,X4,X5) = X1 + 2.X2 + X3 + 2.X4 + 2.X5
 || [IF!FAC6220REACH!FAC62202](X1,X2,X3,X4,X5) = 2.X2 + X3 + 2.X4 + 2.X5
 || [REACH](X1,X2,X3,X4) = 2.X1 + X2 + 2.X3 + 2.X4 + 2
 || 
 || Problem 1.3: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  IF!FAC6220REACH!FAC62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,edge(%Z,%U,%W))
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%U,%Y,union(%V,%W),empty)
 ||  IF!FAC6220REACH!FAC62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,%W)
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Strongly Connected Components:
 || ->->Cycle:
 || ->->-> Pairs:
 ||  IF!FAC6220REACH!FAC62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,edge(%Z,%U,%W))
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 || ->->-> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || 
 || Problem 1.3: 
 || 
 || Subterm Processor:
 || -> Pairs:
 ||  IF!FAC6220REACH!FAC62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> REACH(%X,%Y,%V,edge(%Z,%U,%W))
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Projection:
 ||  pi(IF!FAC6220REACH!FAC62201) = 4
 ||  pi(REACH) = 3
 || 
 || Problem 1.3: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  REACH(%X,%Y,edge(%Z,%U,%V),%W) -> IF!FAC6220REACH!FAC62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 || -> Rules:
 ||  eq(0,0) -> true
 ||  eq(0,s(%X)) -> false
 ||  eq(s(%X),0) -> false
 ||  eq(s(%X),s(%Y)) -> eq(%X,%Y)
 ||  if!fac6220reach!fac62201(false,%X,%Y,edge(%Z,%U,%V),%W) -> reach(%X,%Y,%V,edge(%Z,%U,%W))
 ||  if!fac6220reach!fac62201(true,%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62202(eq(%Y,%U),%X,%Y,edge(%Z,%U,%V),%W)
 ||  if!fac6220reach!fac62202(false,%X,%Y,edge(%Z,%U,%V),%W) -> or(reach(%X,%Y,%V,%W),reach(%U,%Y,union(%V,%W),empty))
 ||  if!fac6220reach!fac62202(true,%X,%Y,edge(%Z,%U,%V),%W) -> true
 ||  or(false,%X) -> %X
 ||  or(true,%X) -> true
 ||  reach(%X,%Y,edge(%Z,%U,%V),%W) -> if!fac6220reach!fac62201(eq(%X,%Z),%X,%Y,edge(%Z,%U,%V),%W)
 ||  reach(%X,%Y,empty,%Z) -> false
 ||  union(edge(%X,%Y,%Z),%U) -> edge(%X,%Y,union(%Z,%U))
 ||  union(empty,%X) -> %X
 || ->Strongly Connected Components:
 ||  There is no strongly connected component
 || 
 || The problem is finite.
 || 
We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]).

We thus obtain the following dependency pair problem (P_0, R_0, computable, formative):

  Dependency Pairs P_0:

    0] map#(F, cons(X, Y)) =#> map#(F, Y)   
    1] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    2] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    3] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    eq(0, 0) => true 
    eq(0, s(X)) => false 
    eq(s(X), 0) => false 
    eq(s(X), s(Y)) => eq(X, Y) 
    or(true, X) => true 
    or(false, X) => X 
    union(empty, X) => X 
    union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) 
    reach(X, Y, empty, Z) => false 
    reach(X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
    if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
    if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) 
    if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true 
    if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 

Thus, the original system is terminating if (P_0, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_0, R_0, computable, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0 
    * 1 : 2, 3 
    * 2 : 1 
    * 3 : 1 

This graph has the following strongly connected components:

  P_1:

    map#(F, cons(X, Y)) =#> map#(F, Y)   

  P_2:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(F, Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_2, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_2, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(filter2#) = 4 
  nu(filter#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by (P_3, R_0, computable, f), where P_3 contains:

  filter2#(true, F, X, Y) =#> filter#(F, Y)   
  filter2#(false, F, X, Y) =#> filter#(F, Y)   

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_3, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_3, R_0, computable, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  
    * 1 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

Thus, the original system is terminating if (P_1, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(map#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, computable, f) by ({}, R_0, computable, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.