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Branch-equiv

Basic equivalence relation for branch structures.

Definitions and Theorems

Function: branch-equiv$inline

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

Theorem: branch-equiv-is-an-equivalence

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

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

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

Theorem: branch-fix-under-branch-equiv

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

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

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

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

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

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

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

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

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