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 the external first-order termination prover, this system is indeed terminating:

 || proof of resources/system.trs
 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
 || 
 || 
 || Termination w.r.t. Q of the given QTRS could be proven:
 || 
 || (0) QTRS
 || (1) Overlay + Local Confluence [EQUIVALENT]
 || (2) QTRS
 || (3) DependencyPairsProof [EQUIVALENT]
 || (4) QDP
 || (5) DependencyGraphProof [EQUIVALENT]
 || (6) AND
 ||     (7) QDP
 ||         (8) UsableRulesProof [EQUIVALENT]
 ||         (9) QDP
 ||         (10) QReductionProof [EQUIVALENT]
 ||         (11) QDP
 ||         (12) QDPSizeChangeProof [EQUIVALENT]
 ||         (13) YES
 ||     (14) QDP
 ||         (15) UsableRulesProof [EQUIVALENT]
 ||         (16) QDP
 ||         (17) QReductionProof [EQUIVALENT]
 ||         (18) QDP
 ||         (19) QDPSizeChangeProof [EQUIVALENT]
 ||         (20) YES
 ||     (21) QDP
 ||         (22) UsableRulesProof [EQUIVALENT]
 ||         (23) QDP
 ||         (24) QReductionProof [EQUIVALENT]
 ||         (25) QDP
 ||         (26) QDPOrderProof [EQUIVALENT]
 ||         (27) QDP
 ||         (28) DependencyGraphProof [EQUIVALENT]
 ||         (29) QDP
 ||         (30) UsableRulesProof [EQUIVALENT]
 ||         (31) QDP
 ||         (32) QReductionProof [EQUIVALENT]
 ||         (33) QDP
 ||         (34) QDPSizeChangeProof [EQUIVALENT]
 ||         (35) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following 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))
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) Overlay + Local Confluence (EQUIVALENT)
 || The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following 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))
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || 
 || ----------------------------------------
 || 
 || (3) DependencyPairsProof (EQUIVALENT)
 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
 || ----------------------------------------
 || 
 || (4)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
 ||    UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U)
 ||    REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W)
 ||    REACH(%X, %Y, edge(%Z, %U, %V), %W) -> EQ(%X, %Z)
 ||    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(true, %X, %Y, edge(%Z, %U, %V), %W) -> EQ(%Y, %U)
 ||    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) -> 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(%X, %Y, %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) -> UNION(%V, %W)
 || 
 || The TRS R consists of the following 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))
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (5) DependencyGraphProof (EQUIVALENT)
 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 4 less nodes.
 || ----------------------------------------
 || 
 || (6)
 || Complex Obligation (AND)
 || 
 || ----------------------------------------
 || 
 || (7)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U)
 || 
 || The TRS R consists of the following 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))
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (8) UsableRulesProof (EQUIVALENT)
 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
 || ----------------------------------------
 || 
 || (9)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U)
 || 
 || R is empty.
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (10) QReductionProof (EQUIVALENT)
 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || 
 || ----------------------------------------
 || 
 || (11)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U)
 || 
 || R is empty.
 || Q is empty.
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (12) QDPSizeChangeProof (EQUIVALENT)
 || By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
 || 
 || From the DPs we obtained the following set of size-change graphs:
 || *UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U)
 || The graph contains the following edges 1 > 1, 2 >= 2
 || 
 || 
 || ----------------------------------------
 || 
 || (13)
 || YES
 || 
 || ----------------------------------------
 || 
 || (14)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
 || 
 || The TRS R consists of the following 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))
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (15) UsableRulesProof (EQUIVALENT)
 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
 || ----------------------------------------
 || 
 || (16)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
 || 
 || R is empty.
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (17) QReductionProof (EQUIVALENT)
 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || 
 || ----------------------------------------
 || 
 || (18)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
 || 
 || R is empty.
 || Q is empty.
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (19) QDPSizeChangeProof (EQUIVALENT)
 || By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
 || 
 || From the DPs we obtained the following set of size-change graphs:
 || *EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
 || The graph contains the following edges 1 > 1, 2 > 2
 || 
 || 
 || ----------------------------------------
 || 
 || (20)
 || YES
 || 
 || ----------------------------------------
 || 
 || (21)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    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!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) -> REACH(%U, %Y, union(%V, %W), empty)
 ||    IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W))
 || 
 || The TRS R consists of the following 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))
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (22) UsableRulesProof (EQUIVALENT)
 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
 || ----------------------------------------
 || 
 || (23)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    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!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) -> REACH(%U, %Y, union(%V, %W), empty)
 ||    IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W))
 || 
 || The TRS R consists of the following rules:
 || 
 ||    union(empty, %X) -> %X
 ||    union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U))
 ||    eq(0, 0) -> true
 ||    eq(0, s(%X)) -> false
 ||    eq(s(%X), 0) -> false
 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    or(true, x0)
 ||    or(false, x0)
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (24) QReductionProof (EQUIVALENT)
 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
 || 
 ||    or(true, x0)
 ||    or(false, x0)
 ||    reach(x0, x1, empty, x2)
 ||    reach(x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5)
 ||    if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5)
 || 
 || 
 || ----------------------------------------
 || 
 || (25)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    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!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) -> REACH(%U, %Y, union(%V, %W), empty)
 ||    IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W))
 || 
 || The TRS R consists of the following rules:
 || 
 ||    union(empty, %X) -> %X
 ||    union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U))
 ||    eq(0, 0) -> true
 ||    eq(0, s(%X)) -> false
 ||    eq(s(%X), 0) -> false
 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (26) QDPOrderProof (EQUIVALENT)
 || We use the reduction pair processor [LPAR04,JAR06].
 || 
 || 
 || The following pairs can be oriented strictly and are deleted.
 || 
 ||    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) -> REACH(%U, %Y, union(%V, %W), empty)
 || The remaining pairs can at least be oriented weakly.
 || Used ordering:  Polynomial interpretation [POLO]:
 || 
 ||    POL(0) = 0
 ||    POL(IF!FAC6220REACH!FAC62201(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5
 ||    POL(IF!FAC6220REACH!FAC62202(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5
 ||    POL(REACH(x_1, x_2, x_3, x_4)) = x_3 + x_4
 ||    POL(edge(x_1, x_2, x_3)) = 1 + x_3
 ||    POL(empty) = 0
 ||    POL(eq(x_1, x_2)) = 0
 ||    POL(false) = 0
 ||    POL(s(x_1)) = 0
 ||    POL(true) = 0
 ||    POL(union(x_1, x_2)) = x_1 + x_2
 || 
 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
 || 
 ||    union(empty, %X) -> %X
 ||    union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U))
 || 
 || 
 || ----------------------------------------
 || 
 || (27)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    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))
 || 
 || The TRS R consists of the following rules:
 || 
 ||    union(empty, %X) -> %X
 ||    union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U))
 ||    eq(0, 0) -> true
 ||    eq(0, s(%X)) -> false
 ||    eq(s(%X), 0) -> false
 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (28) DependencyGraphProof (EQUIVALENT)
 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
 || ----------------------------------------
 || 
 || (29)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    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)
 || 
 || The TRS R consists of the following rules:
 || 
 ||    union(empty, %X) -> %X
 ||    union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U))
 ||    eq(0, 0) -> true
 ||    eq(0, s(%X)) -> false
 ||    eq(s(%X), 0) -> false
 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (30) UsableRulesProof (EQUIVALENT)
 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
 || ----------------------------------------
 || 
 || (31)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    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)
 || 
 || The TRS R consists of the following rules:
 || 
 ||    eq(0, 0) -> true
 ||    eq(0, s(%X)) -> false
 ||    eq(s(%X), 0) -> false
 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (32) QReductionProof (EQUIVALENT)
 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
 || 
 ||    union(empty, x0)
 ||    union(edge(x0, x1, x2), x3)
 || 
 || 
 || ----------------------------------------
 || 
 || (33)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    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)
 || 
 || The TRS R consists of the following rules:
 || 
 ||    eq(0, 0) -> true
 ||    eq(0, s(%X)) -> false
 ||    eq(s(%X), 0) -> false
 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    eq(0, 0)
 ||    eq(0, s(x0))
 ||    eq(s(x0), 0)
 ||    eq(s(x0), s(x1))
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (34) QDPSizeChangeProof (EQUIVALENT)
 || By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
 || 
 || From the DPs we obtained the following set of size-change graphs:
 || *REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W)
 || The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5
 || 
 || 
 || *IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W))
 || The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3
 || 
 || 
 || ----------------------------------------
 || 
 || (35)
 || YES
 || 
We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] map#(F, cons(X, Y)) =#> F(X)   
    1] map#(F, cons(X, Y)) =#> map#(F, Y)   
    2] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    3] filter#(F, cons(X, Y)) =#> F(X)   
    4] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    5] 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, minimal, formative) is finite.

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

The formative rules of (P_0, R_0) are R_1 ::=

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

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative).

Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

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

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.2 + 2y0 + 2y1 
  edge = \y0y1y2.0 
  empty = 0 
  eq = \y0y1.2 
  false = 0 
  filter = \G0y1.y1 
  filter2 = \y0G1y2y3.2 + 2y2 + 2y3 
  filter2# = \y0G1y2y3.G1(y3) 
  filter# = \G0y1.G0(y1) 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  map = \G0y1.2y1 + 2y1G0(y1) 
  map# = \G0y1.3 + 2G0(y1) 
  or = \y0y1.y1 
  reach = \y0y1y2y3.2y3 
  s = \y0.3 
  true = 0 
  union = \y0y1.y1 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 3 + 2F0(2 + 2x1 + 2x2) > F0(x1) = [[_F0(_x1)]] 
  [[map#(_F0, cons(_x1, _x2))]] = 3 + 2F0(2 + 2x1 + 2x2) >= 3 + 2F0(x2) = [[map#(_F0, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = F0(2 + 2x1 + 2x2) >= F0(x2) = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = F0(2 + 2x1 + 2x2) >= F0(x1) = [[_F0(_x1)]] 
  [[filter2#(true, _F0, _x1, _x2)]] = F0(x2) >= F0(x2) = [[filter#(_F0, _x2)]] 
  [[filter2#(false, _F0, _x1, _x2)]] = F0(x2) >= F0(x2) = [[filter#(_F0, _x2)]] 
  [[eq(0, 0)]] = 2 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 2 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 2 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 2 >= 2 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = x3 >= 0 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 2x2 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 2x5 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, cons(_x1, _x2))]] = 4 + 4x1 + 4x2 + 4x1F0(2 + 2x1 + 2x2) + 4x2F0(2 + 2x1 + 2x2) + 4F0(2 + 2x1 + 2x2) >= 2 + 4x2 + 4x2F0(x2) + 2max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, cons(_x1, _x2))]] = 2 + 2x1 + 2x2 >= 2 + 2x1 + 2x2 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 2 + 2x1 + 2x2 >= 2 + 2x1 + 2x2 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 2 + 2x1 + 2x2 >= x2 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, minimal, formative) by (P_1, R_1, minimal, formative), where P_1 consists of:

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

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_1, minimal, 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 : 3, 4 
    * 2 : 0, 1, 2, 3, 4 
    * 3 : 1, 2 
    * 4 : 1, 2 

This graph has the following strongly connected components:

  P_2:

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

  P_3:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter#(F, cons(X, Y)) =#> F(X)   
    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_1, R_1, m, f) by (P_2, R_1, m, f) and (P_3, R_1, m, f).

Thus, the original system is terminating if each of (P_2, R_1, minimal, formative) and (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

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

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.1 + y0 + y1 
  edge = \y0y1y2.0 
  empty = 0 
  eq = \y0y1.0 
  false = 0 
  filter = \G0y1.1 + 2y1 
  filter2 = \y0G1y2y3.2 + y2 + 2y3 
  filter2# = \y0G1y2y3.1 + 2y3 + 2G1(y3) + 3y3G1(y3) 
  filter# = \G0y1.2y1 + 2G0(y1) + 3y1G0(y1) 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  map = \G0y1.y1 + y1G0(y1) 
  or = \y0y1.y1 
  reach = \y0y1y2y3.0 
  s = \y0.3 
  true = 0 
  union = \y0y1.y1 

Using this interpretation, the requirements translate to:

  [[filter#(_F0, cons(_x1, _x2))]] = 2 + 2x1 + 2x2 + 3x1F0(1 + x1 + x2) + 3x2F0(1 + x1 + x2) + 5F0(1 + x1 + x2) > 1 + 2x2 + 2F0(x2) + 3x2F0(x2) = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = 2 + 2x1 + 2x2 + 3x1F0(1 + x1 + x2) + 3x2F0(1 + x1 + x2) + 5F0(1 + x1 + x2) > F0(x1) = [[_F0(_x1)]] 
  [[filter2#(true, _F0, _x1, _x2)]] = 1 + 2x2 + 2F0(x2) + 3x2F0(x2) > 2x2 + 2F0(x2) + 3x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[filter2#(false, _F0, _x1, _x2)]] = 1 + 2x2 + 2F0(x2) + 3x2F0(x2) > 2x2 + 2F0(x2) + 3x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = x3 >= 0 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 0 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, cons(_x1, _x2))]] = 1 + x1 + x2 + F0(1 + x1 + x2) + x1F0(1 + x1 + x2) + x2F0(1 + x1 + x2) >= 1 + x2 + x2F0(x2) + max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, cons(_x1, _x2))]] = 3 + 2x1 + 2x2 >= 2 + x1 + 2x2 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 2 + x1 + 2x2 >= 2 + x1 + 2x2 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 2 + x1 + 2x2 >= 1 + 2x2 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_3, R_1) by ({}, R_1).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_1, minimal, 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 [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_1, minimal, f) by ({}, R_1, minimal, 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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.