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Flatten-res

Flatten-res-equiv

Basic equivalence relation for flatten-res structures.

Definitions and Theorems

Function: flatten-res-equiv$inline

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

Theorem: flatten-res-equiv-is-an-equivalence

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

Theorem: flatten-res-equiv-implies-equal-flatten-res-fix-1

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

Theorem: flatten-res-fix-under-flatten-res-equiv

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

Theorem: equal-of-flatten-res-fix-1-forward-to-flatten-res-equiv

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

Theorem: equal-of-flatten-res-fix-2-forward-to-flatten-res-equiv

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

Theorem: flatten-res-equiv-of-flatten-res-fix-1-forward

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

Theorem: flatten-res-equiv-of-flatten-res-fix-2-forward

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