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• Omaps

Map

A fixtype of omaps.

This is similar to the fixtype set::set of osets.

Definitions and Theorems

Function: mequiv$inline

(defun mequiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (mapp acl2::x) (mapp acl2::y))))
  (equal (mfix acl2::x) (mfix acl2::y)))

Theorem: mequiv-is-an-equivalence

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

Theorem: mequiv-implies-equal-mfix-1

(defthm mequiv-implies-equal-mfix-1
  (implies (mequiv acl2::x x-equiv)
           (equal (mfix acl2::x) (mfix x-equiv)))
  :rule-classes (:congruence))

Theorem: mfix-under-mequiv

(defthm mfix-under-mequiv
  (mequiv (mfix acl2::x) acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-mfix-1-forward-to-mequiv

(defthm equal-of-mfix-1-forward-to-mequiv
  (implies (equal (mfix acl2::x) acl2::y)
           (mequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-mfix-2-forward-to-mequiv

(defthm equal-of-mfix-2-forward-to-mequiv
  (implies (equal acl2::x (mfix acl2::y))
           (mequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: mequiv-of-mfix-1-forward

(defthm mequiv-of-mfix-1-forward
  (implies (mequiv (mfix acl2::x) acl2::y)
           (mequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: mequiv-of-mfix-2-forward

(defthm mequiv-of-mfix-2-forward
  (implies (mequiv acl2::x (mfix acl2::y))
           (mequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)