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Fun-declon

Fun-declon-equiv

Basic equivalence relation for fun-declon structures.

Definitions and Theorems

Function: fun-declon-equiv$inline

(defun fun-declon-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (fun-declonp acl2::x)
                              (fun-declonp acl2::y))))
  (equal (fun-declon-fix acl2::x)
         (fun-declon-fix acl2::y)))

Theorem: fun-declon-equiv-is-an-equivalence

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

Theorem: fun-declon-equiv-implies-equal-fun-declon-fix-1

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

Theorem: fun-declon-fix-under-fun-declon-equiv

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

Theorem: equal-of-fun-declon-fix-1-forward-to-fun-declon-equiv

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

Theorem: equal-of-fun-declon-fix-2-forward-to-fun-declon-equiv

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

Theorem: fun-declon-equiv-of-fun-declon-fix-1-forward

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

Theorem: fun-declon-equiv-of-fun-declon-fix-2-forward

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