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• Programdef

Programdef-equiv

Basic equivalence relation for programdef structures.

Definitions and Theorems

Function: programdef-equiv$inline

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

Theorem: programdef-equiv-is-an-equivalence

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

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

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

Theorem: programdef-fix-under-programdef-equiv

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

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

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

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

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

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

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

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

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