We consider the system Applicative_first_order_05__#3.6.

  Alphabet:

    0 : [] --> b 
    cons : [c * d] --> d 
    false : [] --> a 
    filter : [c -> a * d] --> d 
    filter2 : [a * c -> a * c * d] --> d 
    gcd : [b * b] --> b 
    if!fac6220gcd : [a * b * b] --> b 
    le : [b * b] --> a 
    map : [c -> c * d] --> d 
    minus : [b * b] --> b 
    nil : [] --> d 
    pred : [b] --> b 
    s : [b] --> b 
    true : [] --> a 

  Rules:

    le(0, x) => true 
    le(s(x), 0) => false 
    le(s(x), s(y)) => le(x, y) 
    pred(s(x)) => x 
    minus(x, 0) => x 
    minus(x, s(y)) => pred(minus(x, y)) 
    gcd(0, x) => x 
    gcd(s(x), 0) => s(x) 
    gcd(s(x), s(y)) => if!fac6220gcd(le(y, x), s(x), s(y)) 
    if!fac6220gcd(true, s(x), s(y)) => gcd(minus(x, y), s(y)) 
    if!fac6220gcd(false, s(x), s(y)) => gcd(minus(y, x), s(x)) 
    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:

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  pred(s(X)) => X 
  minus(X, 0) => X 
  minus(X, s(Y)) => pred(minus(X, Y)) 
  gcd(0, X) => X 
  gcd(s(X), 0) => s(X) 
  gcd(s(X), s(Y)) => if!fac6220gcd(le(Y, X), s(X), s(Y)) 
  if!fac6220gcd(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) 
  if!fac6220gcd(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) 

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) TRUE
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 ||    pred(s(%X)) -> %X
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    gcd(0, %X) -> %X
 ||    gcd(s(%X), 0) -> s(%X)
 ||    gcd(s(%X), s(%Y)) -> if!fac6220gcd(le(%Y, %X), s(%X), s(%Y))
 ||    if!fac6220gcd(true, s(%X), s(%Y)) -> gcd(minus(%X, %Y), s(%Y))
 ||    if!fac6220gcd(false, s(%X), s(%Y)) -> gcd(minus(%Y, %X), s(%X))
 || 
 || 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:
 || 
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 ||    pred(s(%X)) -> %X
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    gcd(0, %X) -> %X
 ||    gcd(s(%X), 0) -> s(%X)
 ||    gcd(s(%X), s(%Y)) -> if!fac6220gcd(le(%Y, %X), s(%X), s(%Y))
 ||    if!fac6220gcd(true, s(%X), s(%Y)) -> gcd(minus(%X, %Y), s(%Y))
 ||    if!fac6220gcd(false, s(%X), s(%Y)) -> gcd(minus(%Y, %X), s(%X))
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 
 || ----------------------------------------
 || 
 || (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:
 || 
 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
 ||    MINUS(%X, s(%Y)) -> PRED(minus(%X, %Y))
 ||    MINUS(%X, s(%Y)) -> MINUS(%X, %Y)
 ||    GCD(s(%X), s(%Y)) -> IF!FAC6220GCD(le(%Y, %X), s(%X), s(%Y))
 ||    GCD(s(%X), s(%Y)) -> LE(%Y, %X)
 ||    IF!FAC6220GCD(true, s(%X), s(%Y)) -> GCD(minus(%X, %Y), s(%Y))
 ||    IF!FAC6220GCD(true, s(%X), s(%Y)) -> MINUS(%X, %Y)
 ||    IF!FAC6220GCD(false, s(%X), s(%Y)) -> GCD(minus(%Y, %X), s(%X))
 ||    IF!FAC6220GCD(false, s(%X), s(%Y)) -> MINUS(%Y, %X)
 || 
 || The TRS R consists of the following rules:
 || 
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 ||    pred(s(%X)) -> %X
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    gcd(0, %X) -> %X
 ||    gcd(s(%X), 0) -> s(%X)
 ||    gcd(s(%X), s(%Y)) -> if!fac6220gcd(le(%Y, %X), s(%X), s(%Y))
 ||    if!fac6220gcd(true, s(%X), s(%Y)) -> gcd(minus(%X, %Y), s(%Y))
 ||    if!fac6220gcd(false, s(%X), s(%Y)) -> gcd(minus(%Y, %X), s(%X))
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 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:
 || 
 ||    MINUS(%X, s(%Y)) -> MINUS(%X, %Y)
 || 
 || The TRS R consists of the following rules:
 || 
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 ||    pred(s(%X)) -> %X
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    gcd(0, %X) -> %X
 ||    gcd(s(%X), 0) -> s(%X)
 ||    gcd(s(%X), s(%Y)) -> if!fac6220gcd(le(%Y, %X), s(%X), s(%Y))
 ||    if!fac6220gcd(true, s(%X), s(%Y)) -> gcd(minus(%X, %Y), s(%Y))
 ||    if!fac6220gcd(false, s(%X), s(%Y)) -> gcd(minus(%Y, %X), s(%X))
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 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:
 || 
 ||    MINUS(%X, s(%Y)) -> MINUS(%X, %Y)
 || 
 || R is empty.
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 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].
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 
 || ----------------------------------------
 || 
 || (11)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    MINUS(%X, s(%Y)) -> MINUS(%X, %Y)
 || 
 || 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:
 || *MINUS(%X, s(%Y)) -> MINUS(%X, %Y)
 || 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:
 || 
 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
 || 
 || The TRS R consists of the following rules:
 || 
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 ||    pred(s(%X)) -> %X
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    gcd(0, %X) -> %X
 ||    gcd(s(%X), 0) -> s(%X)
 ||    gcd(s(%X), s(%Y)) -> if!fac6220gcd(le(%Y, %X), s(%X), s(%Y))
 ||    if!fac6220gcd(true, s(%X), s(%Y)) -> gcd(minus(%X, %Y), s(%Y))
 ||    if!fac6220gcd(false, s(%X), s(%Y)) -> gcd(minus(%Y, %X), s(%X))
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 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:
 || 
 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
 || 
 || R is empty.
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 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].
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 
 || ----------------------------------------
 || 
 || (18)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    LE(s(%X), s(%Y)) -> LE(%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:
 || *LE(s(%X), s(%Y)) -> LE(%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:
 || 
 ||    GCD(s(%X), s(%Y)) -> IF!FAC6220GCD(le(%Y, %X), s(%X), s(%Y))
 ||    IF!FAC6220GCD(true, s(%X), s(%Y)) -> GCD(minus(%X, %Y), s(%Y))
 ||    IF!FAC6220GCD(false, s(%X), s(%Y)) -> GCD(minus(%Y, %X), s(%X))
 || 
 || The TRS R consists of the following rules:
 || 
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 ||    pred(s(%X)) -> %X
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    gcd(0, %X) -> %X
 ||    gcd(s(%X), 0) -> s(%X)
 ||    gcd(s(%X), s(%Y)) -> if!fac6220gcd(le(%Y, %X), s(%X), s(%Y))
 ||    if!fac6220gcd(true, s(%X), s(%Y)) -> gcd(minus(%X, %Y), s(%Y))
 ||    if!fac6220gcd(false, s(%X), s(%Y)) -> gcd(minus(%Y, %X), s(%X))
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 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:
 || 
 ||    GCD(s(%X), s(%Y)) -> IF!FAC6220GCD(le(%Y, %X), s(%X), s(%Y))
 ||    IF!FAC6220GCD(true, s(%X), s(%Y)) -> GCD(minus(%X, %Y), s(%Y))
 ||    IF!FAC6220GCD(false, s(%X), s(%Y)) -> GCD(minus(%Y, %X), s(%X))
 || 
 || The TRS R consists of the following rules:
 || 
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    pred(s(%X)) -> %X
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 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].
 || 
 ||    gcd(0, x0)
 ||    gcd(s(x0), 0)
 ||    gcd(s(x0), s(x1))
 ||    if!fac6220gcd(true, s(x0), s(x1))
 ||    if!fac6220gcd(false, s(x0), s(x1))
 || 
 || 
 || ----------------------------------------
 || 
 || (25)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    GCD(s(%X), s(%Y)) -> IF!FAC6220GCD(le(%Y, %X), s(%X), s(%Y))
 ||    IF!FAC6220GCD(true, s(%X), s(%Y)) -> GCD(minus(%X, %Y), s(%Y))
 ||    IF!FAC6220GCD(false, s(%X), s(%Y)) -> GCD(minus(%Y, %X), s(%X))
 || 
 || The TRS R consists of the following rules:
 || 
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    pred(s(%X)) -> %X
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 || 
 || 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!FAC6220GCD(true, s(%X), s(%Y)) -> GCD(minus(%X, %Y), s(%Y))
 ||    IF!FAC6220GCD(false, s(%X), s(%Y)) -> GCD(minus(%Y, %X), s(%X))
 || The remaining pairs can at least be oriented weakly.
 || Used ordering:  Polynomial interpretation [POLO]:
 || 
 ||    POL(0) = 0
 ||    POL(GCD(x_1, x_2)) = 1 + x_1 + x_2
 ||    POL(IF!FAC6220GCD(x_1, x_2, x_3)) = 1 + x_2 + x_3
 ||    POL(false) = 0
 ||    POL(le(x_1, x_2)) = 0
 ||    POL(minus(x_1, x_2)) = x_1
 ||    POL(pred(x_1)) = x_1
 ||    POL(s(x_1)) = 1 + x_1
 ||    POL(true) = 0
 || 
 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
 || 
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    pred(s(%X)) -> %X
 || 
 || 
 || ----------------------------------------
 || 
 || (27)
 || Obligation:
 || Q DP problem:
 || The TRS P consists of the following rules:
 || 
 ||    GCD(s(%X), s(%Y)) -> IF!FAC6220GCD(le(%Y, %X), s(%X), s(%Y))
 || 
 || The TRS R consists of the following rules:
 || 
 ||    minus(%X, 0) -> %X
 ||    minus(%X, s(%Y)) -> pred(minus(%X, %Y))
 ||    pred(s(%X)) -> %X
 ||    le(0, %X) -> true
 ||    le(s(%X), 0) -> false
 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
 || 
 || The set Q consists of the following terms:
 || 
 ||    le(0, x0)
 ||    le(s(x0), 0)
 ||    le(s(x0), s(x1))
 ||    pred(s(x0))
 ||    minus(x0, 0)
 ||    minus(x0, s(x1))
 || 
 || We have to consider all minimal (P,Q,R)-chains.
 || ----------------------------------------
 || 
 || (28) DependencyGraphProof (EQUIVALENT)
 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
 || ----------------------------------------
 || 
 || (29)
 || TRUE
 || 
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:

    le(0, X) => true 
    le(s(X), 0) => false 
    le(s(X), s(Y)) => le(X, Y) 
    pred(s(X)) => X 
    minus(X, 0) => X 
    minus(X, s(Y)) => pred(minus(X, Y)) 
    gcd(0, X) => X 
    gcd(s(X), 0) => s(X) 
    gcd(s(X), s(Y)) => if!fac6220gcd(le(Y, X), s(X), s(Y)) 
    if!fac6220gcd(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) 
    if!fac6220gcd(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) 
    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 ::=

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  pred(s(X)) => X 
  minus(X, 0) => X 
  minus(X, s(Y)) => pred(minus(X, Y)) 
  gcd(0, X) => X 
  gcd(s(X), 0) => s(X) 
  gcd(s(X), s(Y)) => if!fac6220gcd(le(Y, X), s(X), s(Y)) 
  if!fac6220gcd(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) 
  if!fac6220gcd(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) 
  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) 
  le(0, X) >= true 
  le(s(X), 0) >= false 
  le(s(X), s(Y)) >= le(X, Y) 
  pred(s(X)) >= X 
  minus(X, 0) >= X 
  minus(X, s(Y)) >= pred(minus(X, Y)) 
  gcd(0, X) >= X 
  gcd(s(X), 0) >= s(X) 
  gcd(s(X), s(Y)) >= if!fac6220gcd(le(Y, X), s(X), s(Y)) 
  if!fac6220gcd(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) 
  if!fac6220gcd(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) 
  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 + y0 + y1 
  false = 0 
  filter = \G0y1.y1 + y1G0(y1) 
  filter2 = \y0G1y2y3.2 + y2 + y3 + y3G1(y3) 
  filter2# = \y0G1y2y3.y3G1(y3) 
  filter# = \G0y1.y1G0(y1) 
  gcd = \y0y1.y0 + y1 
  if!fac6220gcd = \y0y1y2.y1 + y2 
  le = \y0y1.1 + y0 
  map = \G0y1.y1 + G0(y1) + y1G0(y1) 
  map# = \G0y1.3 + G0(y1) 
  minus = \y0y1.y0 
  pred = \y0.y0 
  s = \y0.2y0 
  true = 0 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 3 + F0(2 + x1 + x2) > F0(x1) = [[_F0(_x1)]] 
  [[map#(_F0, cons(_x1, _x2))]] = 3 + F0(2 + x1 + x2) >= 3 + F0(x2) = [[map#(_F0, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = 2F0(2 + x1 + x2) + x1F0(2 + x1 + x2) + x2F0(2 + x1 + x2) >= x2F0(x2) = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = 2F0(2 + x1 + x2) + x1F0(2 + x1 + x2) + x2F0(2 + x1 + x2) >= F0(x1) = [[_F0(_x1)]] 
  [[filter2#(true, _F0, _x1, _x2)]] = x2F0(x2) >= x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[filter2#(false, _F0, _x1, _x2)]] = x2F0(x2) >= x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[le(0, _x0)]] = 4 >= 0 = [[true]] 
  [[le(s(_x0), 0)]] = 1 + 2x0 >= 0 = [[false]] 
  [[le(s(_x0), s(_x1))]] = 1 + 2x0 >= 1 + x0 = [[le(_x0, _x1)]] 
  [[pred(s(_x0))]] = 2x0 >= x0 = [[_x0]] 
  [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] 
  [[minus(_x0, s(_x1))]] = x0 >= x0 = [[pred(minus(_x0, _x1))]] 
  [[gcd(0, _x0)]] = 3 + x0 >= x0 = [[_x0]] 
  [[gcd(s(_x0), 0)]] = 3 + 2x0 >= 2x0 = [[s(_x0)]] 
  [[gcd(s(_x0), s(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[if!fac6220gcd(le(_x1, _x0), s(_x0), s(_x1))]] 
  [[if!fac6220gcd(true, s(_x0), s(_x1))]] = 2x0 + 2x1 >= x0 + 2x1 = [[gcd(minus(_x0, _x1), s(_x1))]] 
  [[if!fac6220gcd(false, s(_x0), s(_x1))]] = 2x0 + 2x1 >= x1 + 2x0 = [[gcd(minus(_x1, _x0), s(_x0))]] 
  [[map(_F0, cons(_x1, _x2))]] = 2 + x1 + x2 + 3F0(2 + x1 + x2) + x1F0(2 + x1 + x2) + x2F0(2 + x1 + x2) >= 2 + x2 + F0(x2) + x2F0(x2) + max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, cons(_x1, _x2))]] = 2 + x1 + x2 + 2F0(2 + x1 + x2) + x1F0(2 + x1 + x2) + x2F0(2 + x1 + x2) >= 2 + x1 + x2 + x2F0(x2) = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 2 + x1 + x2 + x2F0(x2) >= 2 + x1 + x2 + x2F0(x2) = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 2 + x1 + x2 + x2F0(x2) >= x2 + x2F0(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) 
  le(0, X) >= true 
  le(s(X), 0) >= false 
  le(s(X), s(Y)) >= le(X, Y) 
  pred(s(X)) >= X 
  minus(X, 0) >= X 
  minus(X, s(Y)) >= pred(minus(X, Y)) 
  gcd(0, X) >= X 
  gcd(s(X), 0) >= s(X) 
  gcd(s(X), s(Y)) >= if!fac6220gcd(le(Y, X), s(X), s(Y)) 
  if!fac6220gcd(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) 
  if!fac6220gcd(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) 
  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 
  false = 2 
  filter = \G0y1.y1 
  filter2 = \y0G1y2y3.2 + 2y2 + 2y3 
  filter2# = \y0G1y2y3.y0 + 2y3 + y3G1(y3) 
  filter# = \G0y1.2y1 + y1G0(y1) 
  gcd = \y0y1.y0 + y1 
  if!fac6220gcd = \y0y1y2.y1 + y2 
  le = \y0y1.y0 + 2y1 
  map = \G0y1.y1 + 2y1G0(y1) 
  minus = \y0y1.y0 
  pred = \y0.y0 
  s = \y0.y0 
  true = 0 

Using this interpretation, the requirements translate to:

  [[filter#(_F0, cons(_x1, _x2))]] = 4 + 4x1 + 4x2 + 2x1F0(2 + 2x1 + 2x2) + 2x2F0(2 + 2x1 + 2x2) + 2F0(2 + 2x1 + 2x2) > 2x2 + x2F0(x2) + max(x1, F0(x1)) = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = 4 + 4x1 + 4x2 + 2x1F0(2 + 2x1 + 2x2) + 2x2F0(2 + 2x1 + 2x2) + 2F0(2 + 2x1 + 2x2) > F0(x1) = [[_F0(_x1)]] 
  [[filter2#(true, _F0, _x1, _x2)]] = 2x2 + x2F0(x2) >= 2x2 + x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[filter2#(false, _F0, _x1, _x2)]] = 2 + 2x2 + x2F0(x2) > 2x2 + x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[le(0, _x0)]] = 3 + 2x0 >= 0 = [[true]] 
  [[le(s(_x0), 0)]] = 6 + x0 >= 2 = [[false]] 
  [[le(s(_x0), s(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[le(_x0, _x1)]] 
  [[pred(s(_x0))]] = x0 >= x0 = [[_x0]] 
  [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] 
  [[minus(_x0, s(_x1))]] = x0 >= x0 = [[pred(minus(_x0, _x1))]] 
  [[gcd(0, _x0)]] = 3 + x0 >= x0 = [[_x0]] 
  [[gcd(s(_x0), 0)]] = 3 + x0 >= x0 = [[s(_x0)]] 
  [[gcd(s(_x0), s(_x1))]] = x0 + x1 >= x0 + x1 = [[if!fac6220gcd(le(_x1, _x0), s(_x0), s(_x1))]] 
  [[if!fac6220gcd(true, s(_x0), s(_x1))]] = x0 + x1 >= x0 + x1 = [[gcd(minus(_x0, _x1), s(_x1))]] 
  [[if!fac6220gcd(false, s(_x0), s(_x1))]] = x0 + x1 >= x0 + x1 = [[gcd(minus(_x1, _x0), s(_x0))]] 
  [[map(_F0, cons(_x1, _x2))]] = 2 + 2x1 + 2x2 + 4x1F0(2 + 2x1 + 2x2) + 4x2F0(2 + 2x1 + 2x2) + 4F0(2 + 2x1 + 2x2) >= 2 + 2x2 + 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_3, R_1, minimal, formative) by (P_4, R_1, minimal, formative), where P_4 consists of:

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

Thus, the original system is terminating if each of (P_2, R_1, minimal, formative) and (P_4, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_4, 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 :  

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_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.