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Lhspairs

Lhspairs-equiv

Basic equivalence relation for lhspairs structures.

Definitions and Theorems

Function: lhspairs-equiv$inline

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

Theorem: lhspairs-equiv-is-an-equivalence

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

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

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

Theorem: lhspairs-fix-under-lhspairs-equiv

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

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

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

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

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

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

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

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

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