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• Defstruct-member-info

Defstruct-member-info-equiv

Basic equivalence relation for defstruct-member-info structures.

Definitions and Theorems

Function: defstruct-member-info-equiv$inline

(defun defstruct-member-info-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (defstruct-member-infop acl2::x)
                              (defstruct-member-infop acl2::y))))
  (equal (defstruct-member-info-fix acl2::x)
         (defstruct-member-info-fix acl2::y)))

Theorem: defstruct-member-info-equiv-is-an-equivalence

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

Theorem: defstruct-member-info-equiv-implies-equal-defstruct-member-info-fix-1

(defthm
 defstruct-member-info-equiv-implies-equal-defstruct-member-info-fix-1
 (implies (defstruct-member-info-equiv acl2::x x-equiv)
          (equal (defstruct-member-info-fix acl2::x)
                 (defstruct-member-info-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: defstruct-member-info-fix-under-defstruct-member-info-equiv

(defthm defstruct-member-info-fix-under-defstruct-member-info-equiv
  (defstruct-member-info-equiv (defstruct-member-info-fix acl2::x)
                               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-defstruct-member-info-fix-1-forward-to-defstruct-member-info-equiv

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

Theorem: equal-of-defstruct-member-info-fix-2-forward-to-defstruct-member-info-equiv

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

Theorem: defstruct-member-info-equiv-of-defstruct-member-info-fix-1-forward

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

Theorem: defstruct-member-info-equiv-of-defstruct-member-info-fix-2-forward

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