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Expression

Expression-equiv

Basic equivalence relation for expression structures.

Definitions and Theorems

Function: expression-equiv$inline

(defun expression-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (expressionp acl2::x)
                              (expressionp acl2::y))))
  (equal (expression-fix acl2::x)
         (expression-fix acl2::y)))

Theorem: expression-equiv-is-an-equivalence

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

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

(defthm expression-equiv-implies-equal-expression-fix-1
  (implies (expression-equiv acl2::x x-equiv)
           (equal (expression-fix acl2::x)
                  (expression-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: expression-fix-under-expression-equiv

(defthm expression-fix-under-expression-equiv
  (expression-equiv (expression-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

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

(defthm equal-of-expression-fix-1-forward-to-expression-equiv
  (implies (equal (expression-fix acl2::x) acl2::y)
           (expression-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm equal-of-expression-fix-2-forward-to-expression-equiv
  (implies (equal acl2::x (expression-fix acl2::y))
           (expression-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm expression-equiv-of-expression-fix-1-forward
  (implies (expression-equiv (expression-fix acl2::x)
                             acl2::y)
           (expression-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm expression-equiv-of-expression-fix-2-forward
  (implies (expression-equiv acl2::x (expression-fix acl2::y))
           (expression-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)