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• Sexpr-equivs

4v-sexpr-alist-equiv

X === Y in the sense of sexpr alists if X[k] === Y[k] in the sense of sexprs for all keys k.

This is a universal equivalence, introduced using def-universal-equiv.

Function: 4v-sexpr-alist-equiv

(defun 4v-sexpr-alist-equiv (x y)
  (declare (xargs :non-executable t))
  (declare (xargs :guard t))
  (declare (xargs :non-executable t))
  (prog2$ (throw-nonexec-error '4v-sexpr-alist-equiv
                               (list x y))
          (let ((k (4v-sexpr-alist-equiv-witness x y)))
            (and (iff (hons-assoc-equal k x)
                      (hons-assoc-equal k y))
                 (4v-sexpr-equiv (cdr (hons-assoc-equal k x))
                                 (cdr (hons-assoc-equal k y)))))))

Definitions and Theorems

Theorem: 4v-sexpr-alist-equiv-necc

(defthm 4v-sexpr-alist-equiv-necc
  (implies (not (and (iff (hons-assoc-equal k x)
                          (hons-assoc-equal k y))
                     (4v-sexpr-equiv (cdr (hons-assoc-equal k x))
                                     (cdr (hons-assoc-equal k y)))))
           (not (4v-sexpr-alist-equiv x y))))

Theorem: 4v-sexpr-alist-equiv-witnessing-witness-rule-correct

(defthm 4v-sexpr-alist-equiv-witnessing-witness-rule-correct
 (implies
      (not ((lambda (k y x)
              (not (if (iff (hons-assoc-equal k x)
                            (hons-assoc-equal k y))
                       (4v-sexpr-equiv (cdr (hons-assoc-equal k x))
                                       (cdr (hons-assoc-equal k y)))
                     'nil)))
            (4v-sexpr-alist-equiv-witness x y)
            y x))
      (4v-sexpr-alist-equiv x y))
 :rule-classes nil)

Theorem: 4v-sexpr-alist-equiv-instancing-instance-rule-correct

(defthm 4v-sexpr-alist-equiv-instancing-instance-rule-correct
  (implies (not (if (iff (hons-assoc-equal k x)
                         (hons-assoc-equal k y))
                    (4v-sexpr-equiv (cdr (hons-assoc-equal k x))
                                    (cdr (hons-assoc-equal k y)))
                  'nil))
           (not (4v-sexpr-alist-equiv x y)))
  :rule-classes nil)

Theorem: 4v-sexpr-alist-equiv-is-an-equivalence

(defthm 4v-sexpr-alist-equiv-is-an-equivalence
  (and (booleanp (4v-sexpr-alist-equiv x y))
       (4v-sexpr-alist-equiv x x)
       (implies (4v-sexpr-alist-equiv x y)
                (4v-sexpr-alist-equiv y x))
       (implies (and (4v-sexpr-alist-equiv x y)
                     (4v-sexpr-alist-equiv y z))
                (4v-sexpr-alist-equiv x z)))
  :rule-classes (:equivalence))

Theorem: alist-equiv-refines-4v-sexpr-alist-equiv

(defthm alist-equiv-refines-4v-sexpr-alist-equiv
  (implies (alist-equiv x y)
           (4v-sexpr-alist-equiv x y))
  :rule-classes (:refinement))

Theorem: 4v-sexpr-alist-equiv-refines-keys-equiv

(defthm 4v-sexpr-alist-equiv-refines-keys-equiv
  (implies (4v-sexpr-alist-equiv x y)
           (keys-equiv x y))
  :rule-classes (:refinement))

Theorem: 4v-sexpr-alist-pair-equiv-implies-4v-sexpr-alist-equiv-cons-1

(defthm
      4v-sexpr-alist-pair-equiv-implies-4v-sexpr-alist-equiv-cons-1
  (implies (4v-sexpr-alist-pair-equiv a a-equiv)
           (4v-sexpr-alist-equiv (cons a b)
                                 (cons a-equiv b)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-pair-equiv-hons-assoc-equal-2

(defthm
 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-pair-equiv-hons-assoc-equal-2
 (implies (4v-sexpr-alist-equiv al al-equiv)
          (4v-sexpr-alist-pair-equiv (hons-assoc-equal x al)
                                     (hons-assoc-equal x al-equiv)))
 :rule-classes (:congruence))

Theorem: 4v-sexpr-equiv-cdr-hons-assoc-equal-when-4v-sexpr-alist-equiv

(defthm
      4v-sexpr-equiv-cdr-hons-assoc-equal-when-4v-sexpr-alist-equiv
  (implies (and (4v-sexpr-alist-equiv a b)
                (syntaxp (and (term-order a b)
                              (not (equal a b)))))
           (4v-sexpr-equiv (cdr (hons-assoc-equal k a))
                           (cdr (hons-assoc-equal k b)))))

Theorem: 4v-sexpr-alist-equiv-implies-iff-4v-sexpr-alist-<=-1

(defthm 4v-sexpr-alist-equiv-implies-iff-4v-sexpr-alist-<=-1
  (implies (4v-sexpr-alist-equiv a a-equiv)
           (iff (4v-sexpr-alist-<= a b)
                (4v-sexpr-alist-<= a-equiv b)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-iff-4v-sexpr-alist-<=-2

(defthm 4v-sexpr-alist-equiv-implies-iff-4v-sexpr-alist-<=-2
  (implies (4v-sexpr-alist-equiv b b-equiv)
           (iff (4v-sexpr-alist-<= a b)
                (4v-sexpr-alist-<= a b-equiv)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-equiv-implies-4v-sexpr-alist-equiv-acons-2

(defthm 4v-sexpr-equiv-implies-4v-sexpr-alist-equiv-acons-2
  (implies (4v-sexpr-equiv b b-equiv)
           (4v-sexpr-alist-equiv (acons a b c)
                                 (acons a b-equiv c)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-cons-2

(defthm 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-cons-2
  (implies (4v-sexpr-alist-equiv b b-equiv)
           (4v-sexpr-alist-equiv (cons a b)
                                 (cons a b-equiv)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-append-1

(defthm 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-append-1
  (implies (4v-sexpr-alist-equiv a a-equiv)
           (4v-sexpr-alist-equiv (append a b)
                                 (append a-equiv b)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-append-2

(defthm 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-append-2
  (implies (4v-sexpr-alist-equiv b b-equiv)
           (4v-sexpr-alist-equiv (append a b)
                                 (append a b-equiv)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-equiv-4v-sexpr-restrict-2

(defthm
    4v-sexpr-alist-equiv-implies-4v-sexpr-equiv-4v-sexpr-restrict-2
  (implies (4v-sexpr-alist-equiv al1 al2)
           (4v-sexpr-equiv (4v-sexpr-restrict x al1)
                           (4v-sexpr-restrict x al2)))
  :rule-classes :congruence)

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-list-equiv-4v-sexpr-restrict-list-2

(defthm
 4v-sexpr-alist-equiv-implies-4v-sexpr-list-equiv-4v-sexpr-restrict-list-2
 (implies (4v-sexpr-alist-equiv al1 al2)
          (4v-sexpr-list-equiv (4v-sexpr-restrict-list x al1)
                               (4v-sexpr-restrict-list x al2)))
 :rule-classes :congruence)

Theorem: 4v-sexpr-alist-equiv-implies-4v-env-equiv-4v-sexpr-eval-alist-1

(defthm
    4v-sexpr-alist-equiv-implies-4v-env-equiv-4v-sexpr-eval-alist-1
  (implies (4v-sexpr-alist-equiv al al-equiv)
           (4v-env-equiv (4v-sexpr-eval-alist al env)
                         (4v-sexpr-eval-alist al-equiv env)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-alist-equiv-4v-sexpr-eval-alist-1

(defthm
     4v-sexpr-alist-equiv-implies-alist-equiv-4v-sexpr-eval-alist-1
  (implies (4v-sexpr-alist-equiv al al-equiv)
           (alist-equiv (4v-sexpr-eval-alist al env)
                        (4v-sexpr-eval-alist al-equiv env)))
  :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-compose-alist-1

(defthm
 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-compose-alist-1
 (implies (4v-sexpr-alist-equiv a a-equiv)
          (4v-sexpr-alist-equiv (4v-sexpr-compose-alist a b)
                                (4v-sexpr-compose-alist a-equiv b)))
 :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-alt-necc

(defthm 4v-sexpr-alist-equiv-alt-necc
  (implies (not (and (keys-equiv x y)
                     (4v-env-equiv (4v-sexpr-eval-alist x env)
                                   (4v-sexpr-eval-alist y env))))
           (not (4v-sexpr-alist-equiv-alt x y))))

Theorem: 4v-sexpr-alist-equiv-alt-witnessing-witness-rule-correct

(defthm 4v-sexpr-alist-equiv-alt-witnessing-witness-rule-correct
  (implies
       (not ((lambda (env y x)
               (not (if (keys-equiv x y)
                        (4v-env-equiv (4v-sexpr-eval-alist x env)
                                      (4v-sexpr-eval-alist y env))
                      'nil)))
             (4v-sexpr-alist-equiv-alt-witness x y)
             y x))
       (4v-sexpr-alist-equiv-alt x y))
  :rule-classes nil)

Theorem: 4v-sexpr-alist-equiv-alt-instancing-instance-rule-correct

(defthm 4v-sexpr-alist-equiv-alt-instancing-instance-rule-correct
  (implies (not (if (keys-equiv x y)
                    (4v-env-equiv (4v-sexpr-eval-alist x env)
                                  (4v-sexpr-eval-alist y env))
                  'nil))
           (not (4v-sexpr-alist-equiv-alt x y)))
  :rule-classes nil)

Theorem: 4v-sexpr-alist-equiv-alt-is-an-equivalence

(defthm 4v-sexpr-alist-equiv-alt-is-an-equivalence
  (and (booleanp (4v-sexpr-alist-equiv-alt x y))
       (4v-sexpr-alist-equiv-alt x x)
       (implies (4v-sexpr-alist-equiv-alt x y)
                (4v-sexpr-alist-equiv-alt y x))
       (implies (and (4v-sexpr-alist-equiv-alt x y)
                     (4v-sexpr-alist-equiv-alt y z))
                (4v-sexpr-alist-equiv-alt x z)))
  :rule-classes (:equivalence))

Theorem: 4v-sexpr-alist-equiv-is-alt

(defthm 4v-sexpr-alist-equiv-is-alt
  (iff (4v-sexpr-alist-equiv a b)
       (4v-sexpr-alist-equiv-alt a b)))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-restrict-alist-1

(defthm
 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-restrict-alist-1
 (implies
      (4v-sexpr-alist-equiv a a-equiv)
      (4v-sexpr-alist-equiv (4v-sexpr-restrict-alist a b)
                            (4v-sexpr-restrict-alist a-equiv b)))
 :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-restrict-alist-2

(defthm
 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-restrict-alist-2
 (implies
      (4v-sexpr-alist-equiv b b-equiv)
      (4v-sexpr-alist-equiv (4v-sexpr-restrict-alist a b)
                            (4v-sexpr-restrict-alist a b-equiv)))
 :rule-classes (:congruence))

Theorem: 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-alist-extract-2

(defthm
 4v-sexpr-alist-equiv-implies-4v-sexpr-alist-equiv-4v-sexpr-alist-extract-2
 (implies
      (4v-sexpr-alist-equiv al al-equiv)
      (4v-sexpr-alist-equiv (4v-sexpr-alist-extract keys al)
                            (4v-sexpr-alist-extract keys al-equiv)))
 :rule-classes (:congruence))