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Decl

Decl-equiv

Basic equivalence relation for decl structures.

Definitions and Theorems

Function: decl-equiv$inline

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

Theorem: decl-equiv-is-an-equivalence

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

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

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

Theorem: decl-fix-under-decl-equiv

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

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

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

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

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

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

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

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

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