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Snippet-table

Snippet-table-equiv

Basic equivalence relation for snippet-table structures.

Definitions and Theorems

Function: snippet-table-equiv$inline

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

Theorem: snippet-table-equiv-is-an-equivalence

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

Theorem: snippet-table-equiv-implies-equal-snippet-table-fix-1

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

Theorem: snippet-table-fix-under-snippet-table-equiv

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

Theorem: equal-of-snippet-table-fix-1-forward-to-snippet-table-equiv

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

Theorem: equal-of-snippet-table-fix-2-forward-to-snippet-table-equiv

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

Theorem: snippet-table-equiv-of-snippet-table-fix-1-forward

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

Theorem: snippet-table-equiv-of-snippet-table-fix-2-forward

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