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• Function

Function-equiv

Basic equivalence relation for function structures.

Definitions and Theorems

Function: function-equiv$inline

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

Theorem: function-equiv-is-an-equivalence

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

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

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

Theorem: function-fix-under-function-equiv

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

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

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

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

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

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

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

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

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