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• Scratchlist

Scratchlist-equiv

Basic equivalence relation for scratchlist structures.

Definitions and Theorems

Function: scratchlist-equiv$inline

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

Theorem: scratchlist-equiv-is-an-equivalence

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

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

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

Theorem: scratchlist-fix-under-scratchlist-equiv

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

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

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

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

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

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

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

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

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