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Identifier-option

Identifier-option-equiv

Basic equivalence relation for identifier-option structures.

Definitions and Theorems

Function: identifier-option-equiv$inline

(defun identifier-option-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (identifier-optionp acl2::x)
                              (identifier-optionp acl2::y))))
  (equal (identifier-option-fix acl2::x)
         (identifier-option-fix acl2::y)))

Theorem: identifier-option-equiv-is-an-equivalence

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

Theorem: identifier-option-equiv-implies-equal-identifier-option-fix-1

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

Theorem: identifier-option-fix-under-identifier-option-equiv

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

Theorem: equal-of-identifier-option-fix-1-forward-to-identifier-option-equiv

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

Theorem: equal-of-identifier-option-fix-2-forward-to-identifier-option-equiv

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

Theorem: identifier-option-equiv-of-identifier-option-fix-1-forward

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

Theorem: identifier-option-equiv-of-identifier-option-fix-2-forward

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