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• Deftreeops-table-value

Deftreeops-table-value-equiv

Basic equivalence relation for deftreeops-table-value structures.

Definitions and Theorems

Function: deftreeops-table-value-equiv$inline

(defun deftreeops-table-value-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (deftreeops-table-valuep acl2::x)
                              (deftreeops-table-valuep acl2::y))))
  (equal (deftreeops-table-value-fix acl2::x)
         (deftreeops-table-value-fix acl2::y)))

Theorem: deftreeops-table-value-equiv-is-an-equivalence

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

Theorem: deftreeops-table-value-equiv-implies-equal-deftreeops-table-value-fix-1

(defthm
 deftreeops-table-value-equiv-implies-equal-deftreeops-table-value-fix-1
 (implies (deftreeops-table-value-equiv acl2::x x-equiv)
          (equal (deftreeops-table-value-fix acl2::x)
                 (deftreeops-table-value-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: deftreeops-table-value-fix-under-deftreeops-table-value-equiv

(defthm
      deftreeops-table-value-fix-under-deftreeops-table-value-equiv
  (deftreeops-table-value-equiv (deftreeops-table-value-fix acl2::x)
                                acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-deftreeops-table-value-fix-1-forward-to-deftreeops-table-value-equiv

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

Theorem: equal-of-deftreeops-table-value-fix-2-forward-to-deftreeops-table-value-equiv

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

Theorem: deftreeops-table-value-equiv-of-deftreeops-table-value-fix-1-forward

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

Theorem: deftreeops-table-value-equiv-of-deftreeops-table-value-fix-2-forward

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