Major Section: FREE-VARIABLES-EXAMPLES
The following examples illustrate ACL2's handling of free variables in forward-chaining rules, as well as user control over how such free variables are handled. See free-variables for a background discussion.
; First let us introduce a transitive operation, op, and prove a
; forward-chaining rule stating the transitivity of op.
(encapsulate
(((op * *) => *))
(local (defun op (x y) (< x y)))
(defthm transitivity-of-op
(implies (and (op x y) (op y z)) (op x z))
:rule-classes :forward-chaining))
; The following theorem is proved by forward chaining, using the above rule.
(thm
(implies (and (op u v) (op v w) (op v a))
(op u w)))
; The proof of the theorem just above succeeds because the term (op u v)
; triggers the application of forward-chaining rule transitivity-of-op,
; binding x to u and y to v. Free variable z of that rule is bound to both w
; and to a, resulting in the addition of both (op u w) and (op u a) to the
; context. However, (op v a) happens to be at the front of the context, so
; if only one free-variable binding had been allowed, then z would have only
; been bound to a, not to w, as we now illustrate.
(add-match-free-override :once (:forward-chaining transitivity-of-op))
(thm ; FAILS!
(implies (and (op u v) (op v w) (op v a))
(op u w)))
:ubt! 1
; Starting over, this time we prove transitivity-of-op as a :match-free :once
; forward-chaining rule. Note that the presence of :match-free eliminates
; the free-variables warning that we got the first time.
(encapsulate
(((op * *) => *))
(local (defun op (x y) (< x y)))
(defthm transitivity-of-op
(implies (and (op x y) (op y z)) (op x z))
:rule-classes ((:forward-chaining :match-free :once))))
(thm ; FAILS!
(implies (and (op u v) (op v w) (op v a))
(op u w)))
; Notice that if we swap the order of the last two hypotheses the theorem
; goes through, because this time (op v w) is first in the context.
(thm ; SUCCEEDS!
(implies (and (op u v) (op v a) (op v w))
(op u w)))
:u
; Now let's try setting the default to :once.
(set-match-free-default :once)
; We still get a free-variables warning when we admit this forward-chaining rule.
(encapsulate
(((op * *) => *))
(local (defun op (x y) (< x y)))
(defthm transitivity-of-op
(implies (and (op x y) (op y z)) (op x z))
:rule-classes ((:forward-chaining))))
; This theorem fails--as it should.
(thm ; FAILS!
(implies (and (op u v) (op v w) (op v a))
(op u w)))
; But if we convert this rule (or here, all possible rules) to :all rules,
; then the proof succeeds.
(add-match-free-override :all t)
(thm ; SUCCEEDS!
(implies (and (op u v) (op v w) (op v a))
(op u w)))
; Now let's test a relatively slow :all case (the next thm below).
:ubt! 1
(encapsulate
(((op1 *) => *)
((op3 * * *) => *))
(local (defun op1 (x) (declare (ignore x)) t))
(local (defun op3 (x0 x1 x2)
(declare (ignore x0 x1 x2))
t))
(defthm op1-op3-property
(implies (and (op1 x0) (op1 x1) (op1 x2))
(op3 x0 x1 x2))
:rule-classes ((:forward-chaining :match-free :all))))
; The following succeeds, but takes a little time (about a second in one run).
(thm (implies
(and (op1 a0) (op1 a1) (op1 a2) (op1 a3) (op1 a4) (op1 a5)
(op1 a6) (op1 a7) (op1 a8) (op1 a9) (op1 a10) (op1 a11)
(op1 a12) (op1 a13) (op1 a14) (op1 a15) (op1 a16)
(op1 a17) (op1 a18) (op1 a19) (op1 a20))
(op3 a5 a6 a0)))
(add-match-free-override :once t)
; The same theorem now fails because of the add-match-free-override, but is
; more than an order of magnitude faster.
(thm (implies
(and (op1 a0) (op1 a1) (op1 a2) (op1 a3) (op1 a4) (op1 a5)
(op1 a6) (op1 a7) (op1 a8) (op1 a9) (op1 a10) (op1 a11)
(op1 a12) (op1 a13) (op1 a14) (op1 a15) (op1 a16)
(op1 a17) (op1 a18) (op1 a19) (op1 a20))
(op3 a5 a6 a0)))
; A slight variant succeeds in a negligible amount of time (still with the
; :once override above).
(thm (implies
(and (op1 a0) (op1 a1) (op1 a2) (op1 a3) (op1 a4) (op1 a5)
(op1 a6) (op1 a7) (op1 a8) (op1 a9) (op1 a10) (op1 a11)
(op1 a12) (op1 a13) (op1 a14) (op1 a15) (op1 a16)
(op1 a17) (op1 a18) (op1 a19) (op1 a20))
(op3 a5 a20 a20)))
; Reality check: This shouldn't give a free-variables warning, and everything
; should work great since there are no free variables with this trigger term.
:ubt! 1
(encapsulate
(((op1 *) => *)
((op7 * * * * * * *) => *))
(local (defun op1 (x)
(declare (ignore x))
t))
(local (defun op7 (x0 x1 x2 x3 x4 x5 x6)
(declare (ignore x0 x1 x2 x3 x4 x5 x6))
t))
(defthm op1-op7-property
(implies (and (op1 x0) (op1 x1) (op1 x2)
(op1 x3) (op1 x4) (op1 x5) (op1 x6))
(op7 x0 x1 x2 x3 x4 x5 x6))
:rule-classes ((:forward-chaining
:trigger-terms ((op7 x0 x1 x2 x3 x4 x5 x6))))))
; The following then succeeds, and very quickly.
(thm (implies (and (op1 a0) (op1 a1) (op1 a2)
(op1 a3) (op1 a4) (op1 a5) (op1 a6))
(op7 a4 a6 a5 a6 a6 a6 a6)))
