		Search-engine friendly clone of the ACL2 documentation.

Commit-op

Commit-op-equiv

Basic equivalence relation for commit-op structures.

Definitions and Theorems

Function: commit-op-equiv$inline

(defun commit-op-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (commit-opp acl2::x)
                              (commit-opp acl2::y))))
  (equal (commit-op-fix acl2::x)
         (commit-op-fix acl2::y)))

Theorem: commit-op-equiv-is-an-equivalence

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

Theorem: commit-op-equiv-implies-equal-commit-op-fix-1

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

Theorem: commit-op-fix-under-commit-op-equiv

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

Theorem: equal-of-commit-op-fix-1-forward-to-commit-op-equiv

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

Theorem: equal-of-commit-op-fix-2-forward-to-commit-op-equiv

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

Theorem: commit-op-equiv-of-commit-op-fix-1-forward

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

Theorem: commit-op-equiv-of-commit-op-fix-2-forward

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