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• Sexpr-equivs

4v-sexpr-alist-equiv-alt

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

Function: 4v-sexpr-alist-equiv-alt

(defun 4v-sexpr-alist-equiv-alt (x y)
  (declare (xargs :non-executable t))
  (declare (xargs :guard t))
  (declare (xargs :non-executable t))
  (prog2$ (throw-nonexec-error '4v-sexpr-alist-equiv-alt
                               (list x y))
          (let ((env (4v-sexpr-alist-equiv-alt-witness x y)))
            (and (keys-equiv x y)
                 (4v-env-equiv (4v-sexpr-eval-alist x env)
                               (4v-sexpr-eval-alist y env))))))

Definitions and Theorems

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))