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Validator-state

Validator-state-equiv

Basic equivalence relation for validator-state structures.

Definitions and Theorems

Function: validator-state-equiv$inline

(defun validator-state-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (validator-statep acl2::x)
                              (validator-statep acl2::y))))
  (equal (validator-state-fix acl2::x)
         (validator-state-fix acl2::y)))

Theorem: validator-state-equiv-is-an-equivalence

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

Theorem: validator-state-equiv-implies-equal-validator-state-fix-1

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

Theorem: validator-state-fix-under-validator-state-equiv

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

Theorem: equal-of-validator-state-fix-1-forward-to-validator-state-equiv

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

Theorem: equal-of-validator-state-fix-2-forward-to-validator-state-equiv

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

Theorem: validator-state-equiv-of-validator-state-fix-1-forward

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

Theorem: validator-state-equiv-of-validator-state-fix-2-forward

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