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Lifetime

Lifetime-equiv

Basic equivalence relation for lifetime structures.

Definitions and Theorems

Function: lifetime-equiv$inline

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

Theorem: lifetime-equiv-is-an-equivalence

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

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

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

Theorem: lifetime-fix-under-lifetime-equiv

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

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

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

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

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

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

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

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

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