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Bitarr

Bits-equiv

Bit-for-bit list equivalence: bits-equiv recognizes lists whose nth elements are bit-equiv to one another. It is often useful for reasoning about bit arrays like bitarr.

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

Function: bits-equiv

(defun bits-equiv (x y)
  (declare (xargs :non-executable t))
  (declare (xargs :guard t))
  (prog2$ (throw-nonexec-error 'bits-equiv
                               (list x y))
          (let ((i (bits-equiv-witness x y)))
            (and (bit-equiv (nth i x) (nth i y))))))

Definitions and Theorems

Theorem: bits-equiv-necc

(defthm bits-equiv-necc
  (implies (not (and (bit-equiv (nth i x) (nth i y))))
           (not (bits-equiv x y))))

Theorem: bits-equiv-is-an-equivalence

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

Theorem: bits-equiv-implies-bit-equiv-nth-2

(defthm bits-equiv-implies-bit-equiv-nth-2
  (implies (bits-equiv x x-equiv)
           (bit-equiv (nth i x) (nth i x-equiv)))
  :rule-classes (:congruence))

Theorem: bits-equiv-implies-bits-equiv-update-nth-3

(defthm bits-equiv-implies-bits-equiv-update-nth-3
  (implies (bits-equiv x x-equiv)
           (bits-equiv (update-nth i v x)
                       (update-nth i v x-equiv)))
  :rule-classes (:congruence))

Theorem: bit-equiv-implies-bits-equiv-update-nth-2

(defthm bit-equiv-implies-bits-equiv-update-nth-2
  (implies (bit-equiv v v-equiv)
           (bits-equiv (update-nth i v x)
                       (update-nth i v-equiv x)))
  :rule-classes (:congruence))

Theorem: bits-equiv-implies-bits-equiv-cdr-1

(defthm bits-equiv-implies-bits-equiv-cdr-1
  (implies (bits-equiv a a-equiv)
           (bits-equiv (cdr a) (cdr a-equiv)))
  :rule-classes (:congruence))

Theorem: bits-equiv-implies-bit-equiv-car-1

(defthm bits-equiv-implies-bit-equiv-car-1
  (implies (bits-equiv x x-equiv)
           (bit-equiv (car x) (car x-equiv)))
  :rule-classes (:congruence))

Theorem: bit-equiv-implies-bits-equiv-cons-1

(defthm bit-equiv-implies-bits-equiv-cons-1
  (implies (bit-equiv a a-equiv)
           (bits-equiv (cons a b)
                       (cons a-equiv b)))
  :rule-classes (:congruence))

Theorem: bits-equiv-implies-bits-equiv-cons-2

(defthm bits-equiv-implies-bits-equiv-cons-2
  (implies (bits-equiv b b-equiv)
           (bits-equiv (cons a b)
                       (cons a b-equiv)))
  :rule-classes (:congruence))