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Funcall

Funcall-equiv

Basic equivalence relation for funcall structures.

Definitions and Theorems

Function: funcall-equiv$inline

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

Theorem: funcall-equiv-is-an-equivalence

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

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

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

Theorem: funcall-fix-under-funcall-equiv

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

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

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

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

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

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

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

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

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