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Alias-alist

Alias-alist-equiv

Basic equivalence relation for alias-alist structures.

Definitions and Theorems

Function: alias-alist-equiv$inline

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

Theorem: alias-alist-equiv-is-an-equivalence

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

Theorem: alias-alist-equiv-implies-equal-alias-alist-fix-1

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

Theorem: alias-alist-fix-under-alias-alist-equiv

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

Theorem: equal-of-alias-alist-fix-1-forward-to-alias-alist-equiv

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

Theorem: equal-of-alias-alist-fix-2-forward-to-alias-alist-equiv

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

Theorem: alias-alist-equiv-of-alias-alist-fix-1-forward

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

Theorem: alias-alist-equiv-of-alias-alist-fix-2-forward

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