QUANTIFIERS-USING-DEFUN-SK-EXTENDED

quantification example with details
Major Section:  QUANTIFIERS

See quantifiers-using-defun-sk for the context of this example.

(in-package "ACL2")

; We prove that if every member A of each of two lists satisfies the
; predicate (P A), then this holds of their append; and, conversely.

; Here is a solution using explicit quantification.

(defstub p (x) t)

(defun-sk forall-p (x)
  (forall a (implies (member a x)
                     (p a))))

; The defun-sk above introduces the following axioms.  The idea is that
; (FORALL-P-WITNESS X) picks a counterexample to (forall-p x) if there is one.

;   (DEFUN FORALL-P (X)
;     (LET ((A (FORALL-P-WITNESS X)))
;          (IMPLIES (MEMBER A X) (P A))))
; 
;   (DEFTHM FORALL-P-NECC
;     (IMPLIES (NOT (IMPLIES (MEMBER A X) (P A)))
;              (NOT (FORALL-P X)))
;     :HINTS (("Goal" :USE FORALL-P-WITNESS)))

; The following lemma seems critical.

(defthm member-append
  (iff (member a (append x1 x2))
       (or (member a x1) (member a x2))))

; The proof of forall-p-append seems to go out to lunch, so we break into
; directions as shown below.

(defthm forall-p-append-forward
  (implies (forall-p (append x1 x2))
           (and (forall-p x1) (forall-p x2)))
  :hints (("Goal" ; ``should'' disable forall-p-necc, but no need
           :use
           ((:instance forall-p-necc
                       (x (append x1 x2))
                       (a (forall-p-witness x1)))
            (:instance forall-p-necc
                       (x (append x1 x2))
                       (a (forall-p-witness x2)))))))

(defthm forall-p-append-reverse
  (implies (and (forall-p x1) (forall-p x2))
           (forall-p (append x1 x2)))
  :hints (("Goal"
           :use
           ((:instance forall-p-necc
                       (x x1)
                       (a (forall-p-witness (append x1 x2))))
            (:instance forall-p-necc
                       (x x2)
                       (a (forall-p-witness (append x1 x2))))))))

(defthm forall-p-append
  (equal (forall-p (append x1 x2))
         (and (forall-p x1) (forall-p x2)))
  :hints (("Goal" :use (forall-p-append-forward
                        forall-p-append-reverse))))