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Type

Type-equiv

Basic equivalence relation for type structures.

Definitions and Theorems

Function: type-equiv$inline

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

Theorem: type-equiv-is-an-equivalence

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

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

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

Theorem: type-fix-under-type-equiv

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

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

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

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

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

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

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

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

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