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• Definition

Definition-equiv

Basic equivalence relation for definition structures.

Definitions and Theorems

Function: definition-equiv$inline

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

Theorem: definition-equiv-is-an-equivalence

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

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

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

Theorem: definition-fix-under-definition-equiv

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

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

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

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

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

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

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

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

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