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Ctrex-rule

Ctrex-rule-fix

Fixing function for ctrex-rule structures.

Signature
(ctrex-rule-fix x) → new-x
Arguments: x — Guard (ctrex-rule-p x).
Returns: new-x — Type (ctrex-rule-p new-x).

Definitions and Theorems

Function: ctrex-rule-fix$inline

(defun ctrex-rule-fix$inline (x)
 (declare (xargs :guard (ctrex-rule-p x)))
 (let ((__function__ 'ctrex-rule-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b*
    ((name (acl2::symbol-fix
                (std::prod-car (std::prod-car (std::prod-car x)))))
     (assigned-var
          (pseudo-var-fix
               (std::prod-cdr (std::prod-car (std::prod-car x)))))
     (assign
          (pseudo-term-fix
               (std::prod-car (std::prod-cdr (std::prod-car x)))))
     (match (pseudo-term-subst-fix
                 (std::prod-cdr (std::prod-cdr (std::prod-car x)))))
     (match-conds
          (pseudo-term-subst-fix
               (std::prod-car (std::prod-car (std::prod-cdr x)))))
     (assign-cond
          (pseudo-term-fix
               (std::prod-cdr (std::prod-car (std::prod-cdr x)))))
     (hyp (pseudo-term-fix
               (std::prod-car (std::prod-cdr (std::prod-cdr x)))))
     (equiv
      (pseudo-fnsym-fix
           (std::prod-car
                (std::prod-cdr (std::prod-cdr (std::prod-cdr x))))))
     (ruletype
      (ctrex-ruletype-fix
          (std::prod-cdr
               (std::prod-cdr (std::prod-cdr (std::prod-cdr x)))))))
    (std::prod-cons
        (std::prod-cons (std::prod-cons name assigned-var)
                        (std::prod-cons assign match))
        (std::prod-cons
             (std::prod-cons match-conds assign-cond)
             (std::prod-cons hyp (std::prod-cons equiv ruletype)))))
   :exec x)))

Theorem: ctrex-rule-p-of-ctrex-rule-fix

(defthm ctrex-rule-p-of-ctrex-rule-fix
  (b* ((new-x (ctrex-rule-fix$inline x)))
    (ctrex-rule-p new-x))
  :rule-classes :rewrite)

Theorem: ctrex-rule-fix-when-ctrex-rule-p

(defthm ctrex-rule-fix-when-ctrex-rule-p
  (implies (ctrex-rule-p x)
           (equal (ctrex-rule-fix x) x)))

Function: ctrex-rule-equiv$inline

(defun ctrex-rule-equiv$inline (x y)
  (declare (xargs :guard (and (ctrex-rule-p x)
                              (ctrex-rule-p y))))
  (equal (ctrex-rule-fix x)
         (ctrex-rule-fix y)))

Theorem: ctrex-rule-equiv-is-an-equivalence

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

Theorem: ctrex-rule-equiv-implies-equal-ctrex-rule-fix-1

(defthm ctrex-rule-equiv-implies-equal-ctrex-rule-fix-1
  (implies (ctrex-rule-equiv x x-equiv)
           (equal (ctrex-rule-fix x)
                  (ctrex-rule-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: ctrex-rule-fix-under-ctrex-rule-equiv

(defthm ctrex-rule-fix-under-ctrex-rule-equiv
  (ctrex-rule-equiv (ctrex-rule-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-ctrex-rule-fix-1-forward-to-ctrex-rule-equiv

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

Theorem: equal-of-ctrex-rule-fix-2-forward-to-ctrex-rule-equiv

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

Theorem: ctrex-rule-equiv-of-ctrex-rule-fix-1-forward

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

Theorem: ctrex-rule-equiv-of-ctrex-rule-fix-2-forward

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