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Closure-input

Closure-input-equiv

Basic equivalence relation for closure-input structures.

Definitions and Theorems

Function: closure-input-equiv$inline

(defun closure-input-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (closure-inputp acl2::x)
                              (closure-inputp acl2::y))))
  (equal (closure-input-fix acl2::x)
         (closure-input-fix acl2::y)))

Theorem: closure-input-equiv-is-an-equivalence

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

Theorem: closure-input-equiv-implies-equal-closure-input-fix-1

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

Theorem: closure-input-fix-under-closure-input-equiv

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

Theorem: equal-of-closure-input-fix-1-forward-to-closure-input-equiv

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

Theorem: equal-of-closure-input-fix-2-forward-to-closure-input-equiv

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

Theorem: closure-input-equiv-of-closure-input-fix-1-forward

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

Theorem: closure-input-equiv-of-closure-input-fix-2-forward

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