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• Program

Program-equiv

Basic equivalence relation for program structures.

Definitions and Theorems

Function: program-equiv$inline

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

Theorem: program-equiv-is-an-equivalence

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

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

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

Theorem: program-fix-under-program-equiv

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

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

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

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

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

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

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

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

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