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• Primitive-value

Primitive-value-equiv

Basic equivalence relation for primitive-value structures.

Definitions and Theorems

Function: primitive-value-equiv$inline

(defun primitive-value-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (primitive-valuep acl2::x)
                              (primitive-valuep acl2::y))))
  (equal (primitive-value-fix acl2::x)
         (primitive-value-fix acl2::y)))

Theorem: primitive-value-equiv-is-an-equivalence

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

Theorem: primitive-value-equiv-implies-equal-primitive-value-fix-1

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

Theorem: primitive-value-fix-under-primitive-value-equiv

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

Theorem: equal-of-primitive-value-fix-1-forward-to-primitive-value-equiv

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

Theorem: equal-of-primitive-value-fix-2-forward-to-primitive-value-equiv

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

Theorem: primitive-value-equiv-of-primitive-value-fix-1-forward

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

Theorem: primitive-value-equiv-of-primitive-value-fix-2-forward

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