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• Scratchobj

Scratchobj-equiv

Basic equivalence relation for scratchobj structures.

Definitions and Theorems

Function: scratchobj-equiv$inline

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

Theorem: scratchobj-equiv-is-an-equivalence

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

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

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

Theorem: scratchobj-fix-under-scratchobj-equiv

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

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

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

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

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

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

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

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

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