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Importdecl

Importdecl-equiv

Basic equivalence relation for importdecl structures.

Definitions and Theorems

Function: importdecl-equiv$inline

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

Theorem: importdecl-equiv-is-an-equivalence

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

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

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

Theorem: importdecl-fix-under-importdecl-equiv

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

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

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

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

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

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

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

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

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