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Fhg-args

Fhg-args-equiv

Basic equivalence relation for fhg-args structures.

Definitions and Theorems

Function: fhg-args-equiv$inline

(defun fhg-args-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (fhg-args-p acl2::x)
                              (fhg-args-p acl2::y))))
  (equal (fhg-args-fix acl2::x)
         (fhg-args-fix acl2::y)))

Theorem: fhg-args-equiv-is-an-equivalence

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

Theorem: fhg-args-equiv-implies-equal-fhg-args-fix-1

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

Theorem: fhg-args-fix-under-fhg-args-equiv

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

Theorem: equal-of-fhg-args-fix-1-forward-to-fhg-args-equiv

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

Theorem: equal-of-fhg-args-fix-2-forward-to-fhg-args-equiv

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

Theorem: fhg-args-equiv-of-fhg-args-fix-1-forward

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

Theorem: fhg-args-equiv-of-fhg-args-fix-2-forward

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