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Funparam-equiv

Basic equivalence relation for funparam structures.

Definitions and Theorems

Function: funparam-equiv$inline

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

Theorem: funparam-equiv-is-an-equivalence

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

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

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

Theorem: funparam-fix-under-funparam-equiv

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

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

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

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

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

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

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

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

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