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• Sexpr-equivs

4v-sexpr-list-equiv

X === Y in the sense of sexpr lists if they are equal lengths and Xi evaluates to the same thing as Yi for all i and for all environments.

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

Function: 4v-sexpr-list-equiv

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

Definitions and Theorems

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

(defthm 4v-sexpr-list-equiv-necc
  (implies (not (and (equal (4v-sexpr-eval-list x env)
                            (4v-sexpr-eval-list y env))))
           (not (4v-sexpr-list-equiv x y))))

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

(defthm 4v-sexpr-list-equiv-witnessing-witness-rule-correct
  (implies (not ((lambda (env y x)
                   (not (equal (4v-sexpr-eval-list x env)
                               (4v-sexpr-eval-list y env))))
                 (4v-sexpr-list-equiv-witness x y)
                 y x))
           (4v-sexpr-list-equiv x y))
  :rule-classes nil)

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

(defthm 4v-sexpr-list-equiv-instancing-instance-rule-correct
  (implies (not (equal (4v-sexpr-eval-list x env)
                       (4v-sexpr-eval-list y env)))
           (not (4v-sexpr-list-equiv x y)))
  :rule-classes nil)

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

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

Theorem: 4v-sexpr-list-equiv-alt-def

(defthm 4v-sexpr-list-equiv-alt-def
  (equal (4v-sexpr-list-equiv x y)
         (if (atom x)
             (atom y)
           (and (consp y)
                (4v-sexpr-equiv (car x) (car y))
                (4v-sexpr-list-equiv (cdr x) (cdr y)))))
  :rule-classes :definition)

Theorem: 4v-sexpr-list-equiv-implies-equal-4v-sexpr-eval-list-1

(defthm 4v-sexpr-list-equiv-implies-equal-4v-sexpr-eval-list-1
  (implies (4v-sexpr-list-equiv x x-equiv)
           (equal (4v-sexpr-eval-list x env)
                  (4v-sexpr-eval-list x-equiv env)))
  :rule-classes (:congruence))

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

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

Theorem: 4v-sexpr-equiv-implies-4v-sexpr-list-equiv-cons-1

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

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

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