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• Funinfo+funenv

Funinfo+funenv-equiv

Basic equivalence relation for funinfo+funenv structures.

Definitions and Theorems

Function: funinfo+funenv-equiv$inline

(defun funinfo+funenv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (funinfo+funenv-p acl2::x)
                              (funinfo+funenv-p acl2::y))))
  (equal (funinfo+funenv-fix acl2::x)
         (funinfo+funenv-fix acl2::y)))

Theorem: funinfo+funenv-equiv-is-an-equivalence

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

Theorem: funinfo+funenv-equiv-implies-equal-funinfo+funenv-fix-1

(defthm funinfo+funenv-equiv-implies-equal-funinfo+funenv-fix-1
  (implies (funinfo+funenv-equiv acl2::x x-equiv)
           (equal (funinfo+funenv-fix acl2::x)
                  (funinfo+funenv-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: funinfo+funenv-fix-under-funinfo+funenv-equiv

(defthm funinfo+funenv-fix-under-funinfo+funenv-equiv
  (funinfo+funenv-equiv (funinfo+funenv-fix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-funinfo+funenv-fix-1-forward-to-funinfo+funenv-equiv

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

Theorem: equal-of-funinfo+funenv-fix-2-forward-to-funinfo+funenv-equiv

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

Theorem: funinfo+funenv-equiv-of-funinfo+funenv-fix-1-forward

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

Theorem: funinfo+funenv-equiv-of-funinfo+funenv-fix-2-forward

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