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Initializer

Initializer-equiv

Basic equivalence relation for initializer structures.

Definitions and Theorems

Function: initializer-equiv$inline

(defun initializer-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (initializerp acl2::x)
                              (initializerp acl2::y))))
  (equal (initializer-fix acl2::x)
         (initializer-fix acl2::y)))

Theorem: initializer-equiv-is-an-equivalence

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

Theorem: initializer-equiv-implies-equal-initializer-fix-1

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

Theorem: initializer-fix-under-initializer-equiv

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

Theorem: equal-of-initializer-fix-1-forward-to-initializer-equiv

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

Theorem: equal-of-initializer-fix-2-forward-to-initializer-equiv

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

Theorem: initializer-equiv-of-initializer-fix-1-forward

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

Theorem: initializer-equiv-of-initializer-fix-2-forward

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