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Initdeclor

Initdeclor-equiv

Basic equivalence relation for initdeclor structures.

Definitions and Theorems

Function: initdeclor-equiv$inline

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

Theorem: initdeclor-equiv-is-an-equivalence

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

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

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

Theorem: initdeclor-fix-under-initdeclor-equiv

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

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

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

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

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

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

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

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

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