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• Vl-sigma

Vl-sigma-equiv

Basic equivalence relation for vl-sigma structures.

Definitions and Theorems

Function: vl-sigma-equiv$inline

(defun vl-sigma-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (vl-sigma-p acl2::x)
                              (vl-sigma-p acl2::y))))
  (equal (vl-sigma-fix acl2::x)
         (vl-sigma-fix acl2::y)))

Theorem: vl-sigma-equiv-is-an-equivalence

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

Theorem: vl-sigma-equiv-implies-equal-vl-sigma-fix-1

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

Theorem: vl-sigma-fix-under-vl-sigma-equiv

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

Theorem: equal-of-vl-sigma-fix-1-forward-to-vl-sigma-equiv

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

Theorem: equal-of-vl-sigma-fix-2-forward-to-vl-sigma-equiv

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

Theorem: vl-sigma-equiv-of-vl-sigma-fix-1-forward

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

Theorem: vl-sigma-equiv-of-vl-sigma-fix-2-forward

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