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Smtlink-process-user-hint

Hypothesis-lst-syntax

Definitions and Theorems

Function: hypothesis-syntax-p

(defun hypothesis-syntax-p (term)
  (declare (xargs :guard t))
  (let ((acl2::__function__ 'hypothesis-syntax-p))
    (declare (ignorable acl2::__function__))
    (or (and (atom term) (equal term nil))
        (and (true-listp term)
             (car term)
             (not (cdr term))
             (pseudo-termp (car term)))
        (and (true-listp term)
             (car term)
             (cadr term)
             (not (cdddr term))
             (pseudo-termp (car term))
             (equal (cadr term) ':hints)
             (hints-syntax-p (caddr term))))))

Theorem: booleanp-of-hypothesis-syntax-p

(defthm booleanp-of-hypothesis-syntax-p
  (b* ((syntax-good? (hypothesis-syntax-p term)))
    (booleanp syntax-good?))
  :rule-classes :rewrite)

Theorem: true-listp-of-caddr

(defthm true-listp-of-caddr
  (implies (and (consp term)
                (consp (cdr term))
                (true-listp (cddr term))
                (equal (+ 2 (len (cddr term))) 3)
                (pseudo-termp (car term))
                (equal (cadr term) :hints)
                (hints-syntax-p (caddr term)))
           (true-listp (caddr term))))

Function: hypothesis-syntax-fix

(defun hypothesis-syntax-fix (term)
  (declare (xargs :guard (hypothesis-syntax-p term)))
  (let ((acl2::__function__ 'hypothesis-syntax-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic (if (hypothesis-syntax-p term) term nil)
         :exec term)))

Theorem: hypothesis-syntax-p-of-hypothesis-syntax-fix

(defthm hypothesis-syntax-p-of-hypothesis-syntax-fix
  (b* ((fixed-term (hypothesis-syntax-fix term)))
    (hypothesis-syntax-p fixed-term))
  :rule-classes :rewrite)

Function: hypothesis-syntax-equiv$inline

(defun hypothesis-syntax-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (hypothesis-syntax-p acl2::x)
                              (hypothesis-syntax-p acl2::y))))
  (equal (hypothesis-syntax-fix acl2::x)
         (hypothesis-syntax-fix acl2::y)))

Theorem: hypothesis-syntax-equiv-is-an-equivalence

(defthm hypothesis-syntax-equiv-is-an-equivalence
  (and (booleanp (hypothesis-syntax-equiv x y))
       (hypothesis-syntax-equiv x x)
       (implies (hypothesis-syntax-equiv x y)
                (hypothesis-syntax-equiv y x))
       (implies (and (hypothesis-syntax-equiv x y)
                     (hypothesis-syntax-equiv y z))
                (hypothesis-syntax-equiv x z)))
  :rule-classes (:equivalence))

Theorem: hypothesis-syntax-equiv-implies-equal-hypothesis-syntax-fix-1

(defthm
      hypothesis-syntax-equiv-implies-equal-hypothesis-syntax-fix-1
  (implies (hypothesis-syntax-equiv acl2::x x-equiv)
           (equal (hypothesis-syntax-fix acl2::x)
                  (hypothesis-syntax-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: hypothesis-syntax-fix-under-hypothesis-syntax-equiv

(defthm hypothesis-syntax-fix-under-hypothesis-syntax-equiv
  (hypothesis-syntax-equiv (hypothesis-syntax-fix acl2::x)
                           acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-hypothesis-syntax-fix-1-forward-to-hypothesis-syntax-equiv

(defthm
 equal-of-hypothesis-syntax-fix-1-forward-to-hypothesis-syntax-equiv
 (implies (equal (hypothesis-syntax-fix acl2::x)
                 acl2::y)
          (hypothesis-syntax-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-hypothesis-syntax-fix-2-forward-to-hypothesis-syntax-equiv

(defthm
 equal-of-hypothesis-syntax-fix-2-forward-to-hypothesis-syntax-equiv
 (implies (equal acl2::x (hypothesis-syntax-fix acl2::y))
          (hypothesis-syntax-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: hypothesis-syntax-equiv-of-hypothesis-syntax-fix-1-forward

(defthm hypothesis-syntax-equiv-of-hypothesis-syntax-fix-1-forward
  (implies (hypothesis-syntax-equiv (hypothesis-syntax-fix acl2::x)
                                    acl2::y)
           (hypothesis-syntax-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: hypothesis-syntax-equiv-of-hypothesis-syntax-fix-2-forward

(defthm hypothesis-syntax-equiv-of-hypothesis-syntax-fix-2-forward
 (implies
   (hypothesis-syntax-equiv acl2::x (hypothesis-syntax-fix acl2::y))
   (hypothesis-syntax-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Function: hypothesis-lst-syntax-p

(defun hypothesis-lst-syntax-p (term)
  (declare (xargs :guard t))
  (let ((acl2::__function__ 'hypothesis-lst-syntax-p))
    (declare (ignorable acl2::__function__))
    (b* (((if (atom term)) (equal term nil))
         ((cons first rest) term))
      (and (hypothesis-syntax-p first)
           (hypothesis-lst-syntax-p rest)))))

Theorem: booleanp-of-hypothesis-lst-syntax-p

(defthm booleanp-of-hypothesis-lst-syntax-p
  (b* ((syntax-good? (hypothesis-lst-syntax-p term)))
    (booleanp syntax-good?))
  :rule-classes :rewrite)

Function: hypothesis-lst-syntax-fix

(defun hypothesis-lst-syntax-fix (term)
  (declare (xargs :guard (hypothesis-lst-syntax-p term)))
  (let ((acl2::__function__ 'hypothesis-lst-syntax-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (consp term)
             (cons (hypothesis-syntax-fix (car term))
                   (hypothesis-lst-syntax-fix (cdr term)))
           nil)
         :exec term)))

Theorem: hypothesis-lst-syntax-p-of-hypothesis-lst-syntax-fix

(defthm hypothesis-lst-syntax-p-of-hypothesis-lst-syntax-fix
  (b* ((fixed-term (hypothesis-lst-syntax-fix term)))
    (hypothesis-lst-syntax-p fixed-term))
  :rule-classes :rewrite)

Function: hypothesis-lst-syntax-equiv$inline

(defun hypothesis-lst-syntax-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (hypothesis-lst-syntax-p acl2::x)
                              (hypothesis-lst-syntax-p acl2::y))))
  (equal (hypothesis-lst-syntax-fix acl2::x)
         (hypothesis-lst-syntax-fix acl2::y)))

Theorem: hypothesis-lst-syntax-equiv-is-an-equivalence

(defthm hypothesis-lst-syntax-equiv-is-an-equivalence
  (and (booleanp (hypothesis-lst-syntax-equiv x y))
       (hypothesis-lst-syntax-equiv x x)
       (implies (hypothesis-lst-syntax-equiv x y)
                (hypothesis-lst-syntax-equiv y x))
       (implies (and (hypothesis-lst-syntax-equiv x y)
                     (hypothesis-lst-syntax-equiv y z))
                (hypothesis-lst-syntax-equiv x z)))
  :rule-classes (:equivalence))

Theorem: hypothesis-lst-syntax-equiv-implies-equal-hypothesis-lst-syntax-fix-1

(defthm
 hypothesis-lst-syntax-equiv-implies-equal-hypothesis-lst-syntax-fix-1
 (implies (hypothesis-lst-syntax-equiv acl2::x x-equiv)
          (equal (hypothesis-lst-syntax-fix acl2::x)
                 (hypothesis-lst-syntax-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: hypothesis-lst-syntax-fix-under-hypothesis-lst-syntax-equiv

(defthm hypothesis-lst-syntax-fix-under-hypothesis-lst-syntax-equiv
  (hypothesis-lst-syntax-equiv (hypothesis-lst-syntax-fix acl2::x)
                               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-hypothesis-lst-syntax-fix-1-forward-to-hypothesis-lst-syntax-equiv

(defthm
 equal-of-hypothesis-lst-syntax-fix-1-forward-to-hypothesis-lst-syntax-equiv
 (implies (equal (hypothesis-lst-syntax-fix acl2::x)
                 acl2::y)
          (hypothesis-lst-syntax-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-hypothesis-lst-syntax-fix-2-forward-to-hypothesis-lst-syntax-equiv

(defthm
 equal-of-hypothesis-lst-syntax-fix-2-forward-to-hypothesis-lst-syntax-equiv
 (implies (equal acl2::x
                 (hypothesis-lst-syntax-fix acl2::y))
          (hypothesis-lst-syntax-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: hypothesis-lst-syntax-equiv-of-hypothesis-lst-syntax-fix-1-forward

(defthm
 hypothesis-lst-syntax-equiv-of-hypothesis-lst-syntax-fix-1-forward
 (implies
    (hypothesis-lst-syntax-equiv (hypothesis-lst-syntax-fix acl2::x)
                                 acl2::y)
    (hypothesis-lst-syntax-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: hypothesis-lst-syntax-equiv-of-hypothesis-lst-syntax-fix-2-forward

(defthm
 hypothesis-lst-syntax-equiv-of-hypothesis-lst-syntax-fix-2-forward
 (implies
   (hypothesis-lst-syntax-equiv acl2::x
                                (hypothesis-lst-syntax-fix acl2::y))
   (hypothesis-lst-syntax-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Subtopics

Hypothesis-syntax-p: Recognizer for hypothesis-syntax.
Hypothesis-lst-syntax-p: Recognizer for hypothesis-lst-syntax.
Hypothesis-lst-syntax-fix: Fixing function for a hypothesis-lst-syntax-p.
Hypothesis-syntax-fix: Fixing function for a hypothesis-syntax-p.