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Command

Command-equiv

Basic equivalence relation for command structures.

Definitions and Theorems

Function: command-equiv$inline

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

Theorem: command-equiv-is-an-equivalence

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

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

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

Theorem: command-fix-under-command-equiv

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

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

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

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

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

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

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

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

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