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Bits-as-digits-in-base-2

Nat=>bebits

Convert a natural number to its big-endian list of bits, seen as digits in base 2, of specified length.

Signature
(nat=>bebits width nat) → digits
Arguments: width — Guard (natp width).; nat — Guard (natp nat).
Returns: digits — Type (bit-listp digits).

Definitions and Theorems

Function: nat=>bebits

(defun nat=>bebits (width nat)
  (declare (xargs :guard (and (natp width) (natp nat))))
  (declare (xargs :guard (< nat (expt 2 width))))
  (let ((__function__ 'nat=>bebits))
    (declare (ignorable __function__))
    (nat=>bendian 2 width nat)))

Theorem: bit-listp-of-nat=>bebits

(defthm bit-listp-of-nat=>bebits
  (b* ((digits (nat=>bebits width nat)))
    (bit-listp digits))
  :rule-classes :rewrite)

Theorem: nat=>bebits-of-nfix-width

(defthm nat=>bebits-of-nfix-width
  (equal (nat=>bebits (nfix width) nat)
         (nat=>bebits width nat)))

Theorem: nat=>bebits-nat-equiv-congruence-on-width

(defthm nat=>bebits-nat-equiv-congruence-on-width
  (implies (nat-equiv width width-equiv)
           (equal (nat=>bebits width nat)
                  (nat=>bebits width-equiv nat)))
  :rule-classes :congruence)

Theorem: nat=>bebits-of-nfix-nat

(defthm nat=>bebits-of-nfix-nat
  (equal (nat=>bebits width (nfix nat))
         (nat=>bebits width nat)))

Theorem: nat=>bebits-nat-equiv-congruence-on-nat

(defthm nat=>bebits-nat-equiv-congruence-on-nat
  (implies (nat-equiv nat nat-equiv)
           (equal (nat=>bebits width nat)
                  (nat=>bebits width nat-equiv)))
  :rule-classes :congruence)

Theorem: len-of-nat=>bebits

(defthm len-of-nat=>bebits
  (equal (len (nat=>bebits width nat))
         (nfix width)))

Subtopics

Nat=>bebits-injectivity
Nat=>bebits-of-bebits=>nat
Bebits=>nat-of-nat=>bebits