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• Proposal

Proposal-equiv

Basic equivalence relation for proposal structures.

Definitions and Theorems

Function: proposal-equiv$inline

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

Theorem: proposal-equiv-is-an-equivalence

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

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

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

Theorem: proposal-fix-under-proposal-equiv

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

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

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

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

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

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

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

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

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