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• Ident-ident-alist

Ident-ident-alist-equiv

Basic equivalence relation for ident-ident-alist structures.

Definitions and Theorems

Function: ident-ident-alist-equiv$inline

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

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

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

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

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

Theorem: ident-ident-alist-fix-under-ident-ident-alist-equiv

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

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

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

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

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

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

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

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

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