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• Integer-literal

Integer-literal-equiv

Basic equivalence relation for integer-literal structures.

Definitions and Theorems

Function: integer-literal-equiv$inline

(defun integer-literal-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (integer-literalp acl2::x)
                              (integer-literalp acl2::y))))
  (equal (integer-literal-fix acl2::x)
         (integer-literal-fix acl2::y)))

Theorem: integer-literal-equiv-is-an-equivalence

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

Theorem: integer-literal-equiv-implies-equal-integer-literal-fix-1

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

Theorem: integer-literal-fix-under-integer-literal-equiv

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

Theorem: equal-of-integer-literal-fix-1-forward-to-integer-literal-equiv

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

Theorem: equal-of-integer-literal-fix-2-forward-to-integer-literal-equiv

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

Theorem: integer-literal-equiv-of-integer-literal-fix-1-forward

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

Theorem: integer-literal-equiv-of-integer-literal-fix-2-forward

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