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Lhslist

Lhslist-equiv

Basic equivalence relation for lhslist structures.

Definitions and Theorems

Function: lhslist-equiv$inline

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

Theorem: lhslist-equiv-is-an-equivalence

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

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

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

Theorem: lhslist-fix-under-lhslist-equiv

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

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

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

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

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

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

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

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

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