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Pinst

Pinst-equiv

Basic equivalence relation for pinst structures.

Definitions and Theorems

Function: pinst-equiv$inline

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

Theorem: pinst-equiv-is-an-equivalence

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

Theorem: pinst-equiv-implies-equal-pinst-fix-1

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

Theorem: pinst-fix-under-pinst-equiv

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

Theorem: equal-of-pinst-fix-1-forward-to-pinst-equiv

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

Theorem: equal-of-pinst-fix-2-forward-to-pinst-equiv

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

Theorem: pinst-equiv-of-pinst-fix-1-forward

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

Theorem: pinst-equiv-of-pinst-fix-2-forward

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