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  • Math
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• Section-info

Section-info-equiv

Basic equivalence relation for section-info structures.

Definitions and Theorems

Function: section-info-equiv$inline

(defun section-info-equiv$inline (x y)
  (declare (xargs :guard (and (section-info-p x)
                              (section-info-p y))))
  (equal (section-info-fix x)
         (section-info-fix y)))

Theorem: section-info-equiv-is-an-equivalence

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

Theorem: section-info-equiv-implies-equal-section-info-fix-1

(defthm section-info-equiv-implies-equal-section-info-fix-1
  (implies (section-info-equiv x x-equiv)
           (equal (section-info-fix x)
                  (section-info-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: section-info-fix-under-section-info-equiv

(defthm section-info-fix-under-section-info-equiv
  (section-info-equiv (section-info-fix x)
                      x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-section-info-fix-1-forward-to-section-info-equiv

(defthm equal-of-section-info-fix-1-forward-to-section-info-equiv
  (implies (equal (section-info-fix x) y)
           (section-info-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-section-info-fix-2-forward-to-section-info-equiv

(defthm equal-of-section-info-fix-2-forward-to-section-info-equiv
  (implies (equal x (section-info-fix y))
           (section-info-equiv x y))
  :rule-classes :forward-chaining)

Theorem: section-info-equiv-of-section-info-fix-1-forward

(defthm section-info-equiv-of-section-info-fix-1-forward
  (implies (section-info-equiv (section-info-fix x)
                               y)
           (section-info-equiv x y))
  :rule-classes :forward-chaining)

Theorem: section-info-equiv-of-section-info-fix-2-forward

(defthm section-info-equiv-of-section-info-fix-2-forward
  (implies (section-info-equiv x (section-info-fix y))
           (section-info-equiv x y))
  :rule-classes :forward-chaining)