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• Smtlink-option-name-lst

Smtlink-option-name-lst-equiv

Basic equivalence relation for smtlink-option-name-lst structures.

Definitions and Theorems

Function: smtlink-option-name-lst-equiv$inline

(defun smtlink-option-name-lst-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (smtlink-option-name-lst-p acl2::x)
                              (smtlink-option-name-lst-p acl2::y))))
  (equal (smtlink-option-name-lst-fix acl2::x)
         (smtlink-option-name-lst-fix acl2::y)))

Theorem: smtlink-option-name-lst-equiv-is-an-equivalence

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

Theorem: smtlink-option-name-lst-equiv-implies-equal-smtlink-option-name-lst-fix-1

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

Theorem: smtlink-option-name-lst-fix-under-smtlink-option-name-lst-equiv

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

Theorem: equal-of-smtlink-option-name-lst-fix-1-forward-to-smtlink-option-name-lst-equiv

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

Theorem: equal-of-smtlink-option-name-lst-fix-2-forward-to-smtlink-option-name-lst-equiv

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

Theorem: smtlink-option-name-lst-equiv-of-smtlink-option-name-lst-fix-1-forward

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

Theorem: smtlink-option-name-lst-equiv-of-smtlink-option-name-lst-fix-2-forward

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