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• Signed-format

Signed-format-equiv

Basic equivalence relation for signed-format structures.

Definitions and Theorems

Function: signed-format-equiv$inline

(defun signed-format-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (signed-formatp acl2::x)
                              (signed-formatp acl2::y))))
  (equal (signed-format-fix acl2::x)
         (signed-format-fix acl2::y)))

Theorem: signed-format-equiv-is-an-equivalence

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

Theorem: signed-format-equiv-implies-equal-signed-format-fix-1

(defthm signed-format-equiv-implies-equal-signed-format-fix-1
  (implies (signed-format-equiv acl2::x x-equiv)
           (equal (signed-format-fix acl2::x)
                  (signed-format-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: signed-format-fix-under-signed-format-equiv

(defthm signed-format-fix-under-signed-format-equiv
  (signed-format-equiv (signed-format-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-signed-format-fix-1-forward-to-signed-format-equiv

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

Theorem: equal-of-signed-format-fix-2-forward-to-signed-format-equiv

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

Theorem: signed-format-equiv-of-signed-format-fix-1-forward

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

Theorem: signed-format-equiv-of-signed-format-fix-2-forward

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