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Paramdeclor

Paramdeclor-equiv

Basic equivalence relation for paramdeclor structures.

Definitions and Theorems

Function: paramdeclor-equiv$inline

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

Theorem: paramdeclor-equiv-is-an-equivalence

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

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

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

Theorem: paramdeclor-fix-under-paramdeclor-equiv

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

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

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

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

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

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

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

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

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