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• Truth

Index-permute-shrink

Signature

(index-permute-shrink n count mask x numvars) → perm-index

Arguments

n — Guard (natp n).

mask — Guard (natp mask).

x — Guard (natp x).

numvars — Guard (natp numvars).

Returns

perm-index — Type (natp perm-index).

Definitions and Theorems

Function: index-permute-shrink

(defun index-permute-shrink (n count mask x numvars)
  (declare (xargs :guard (and (natp n)
                              (natp mask)
                              (natp x)
                              (natp numvars))))
  (declare
       (xargs :guard (and (<= n numvars)
                          (eql count (logcount (loghead n mask))))))
  (let ((__function__ 'index-permute-shrink))
    (declare (ignorable __function__))
    (b* (((when (mbe :logic (zp (- (nfix numvars) (nfix n)))
                     :exec (eql n numvars)))
          (lnfix x))
         (n (lnfix n))
         (count (mbe :logic (logcount (loghead n (lnfix mask)))
                     :exec count))
         (bit (logbit n (lnfix mask)))
         (x (if (eql bit 1)
                (index-move-down count n x)
              x)))
      (index-permute-shrink (1+ n)
                            (+ bit count)
                            mask x numvars))))

Theorem: natp-of-index-permute-shrink

(defthm natp-of-index-permute-shrink
  (b* ((perm-index (index-permute-shrink n count mask x numvars)))
    (natp perm-index))
  :rule-classes :type-prescription)

Theorem: index-permute-shrink-bound

(defthm index-permute-shrink-bound
  (b* ((?perm-index (index-permute-shrink n count mask x numvars)))
    (implies (< (nfix x) (nfix numvars))
             (< perm-index (nfix numvars))))
  :rule-classes :linear)

Theorem: index-permute-shrink-normalize-count

(defthm index-permute-shrink-normalize-count
  (implies (syntaxp (not (equal count ''nil)))
           (equal (index-permute-shrink n count mask x numvars)
                  (index-permute-shrink n nil mask x numvars))))

Theorem: index-permute-shrink-of-index-permute-stretch

(defthm index-permute-shrink-of-index-permute-stretch
  (equal (index-permute-shrink
              n count mask
              (index-permute-stretch n count2 mask x numvars)
              numvars)
         (nfix x)))

Theorem: index-permute-stretch-of-index-permute-shrink

(defthm index-permute-stretch-of-index-permute-shrink
  (equal (index-permute-stretch
              n count mask
              (index-permute-shrink n count2 mask x numvars)
              numvars)
         (nfix x)))

Theorem: index-permute-stretch-one-to-one

(defthm index-permute-stretch-one-to-one
 (implies
     (not (equal (nfix x) (nfix y)))
     (not (equal (index-permute-stretch n count mask x numvars)
                 (index-permute-stretch n count2 mask y numvars)))))

Theorem: index-permute-shrink-one-to-one

(defthm index-permute-shrink-one-to-one
 (implies
      (not (equal (nfix x) (nfix y)))
      (not (equal (index-permute-shrink n count mask x numvars)
                  (index-permute-shrink n count2 mask y numvars)))))

Theorem: index-permute-shrink-redef

(defthm index-permute-shrink-redef
 (b* ((?perm-index (index-permute-shrink n count mask x numvars)))
  (implies
   (<= (nfix n) (nfix numvars))
   (equal
    perm-index
    (cond
      ((< (nfix x)
          (logcount (loghead n (nfix mask))))
       (nfix x))
      ((< (nfix x) (nfix n))
       (+ (nfix x)
          (- (logcount (loghead numvars (nfix mask)))
             (logcount (loghead n (nfix mask))))))
      ((and (< (nfix x) (nfix numvars))
            (logbitp x (nfix mask)))
       (logcount (loghead x (nfix mask))))
      (t (+ (nfix x)
            (- (logcount (loghead x (loghead numvars (nfix mask)))))
            (logcount (loghead numvars (nfix mask))))))))))

Theorem: index-permute-shrink-redef-base

(defthm index-permute-shrink-redef-base
  (equal (index-permute-shrink 0 count mask x numvars)
         (b* ((x (nfix x))
              (numvars (nfix numvars))
              (mask (nfix mask)))
           (cond ((and (< x numvars) (logbitp x mask))
                  (logcount (loghead x mask)))
                 ((<= numvars x) x)
                 (t (+ (logcount (loghead x (lognot mask)))
                       (logcount (loghead numvars mask))))))))

Theorem: index-permute-shrink-of-nfix-n

(defthm index-permute-shrink-of-nfix-n
  (equal (index-permute-shrink (nfix n)
                               count mask x numvars)
         (index-permute-shrink n count mask x numvars)))

Theorem: index-permute-shrink-nat-equiv-congruence-on-n

(defthm index-permute-shrink-nat-equiv-congruence-on-n
  (implies
       (nat-equiv n n-equiv)
       (equal (index-permute-shrink n count mask x numvars)
              (index-permute-shrink n-equiv count mask x numvars)))
  :rule-classes :congruence)

Theorem: index-permute-shrink-of-nfix-mask

(defthm index-permute-shrink-of-nfix-mask
  (equal (index-permute-shrink n count (nfix mask)
                               x numvars)
         (index-permute-shrink n count mask x numvars)))

Theorem: index-permute-shrink-nat-equiv-congruence-on-mask

(defthm index-permute-shrink-nat-equiv-congruence-on-mask
  (implies
       (nat-equiv mask mask-equiv)
       (equal (index-permute-shrink n count mask x numvars)
              (index-permute-shrink n count mask-equiv x numvars)))
  :rule-classes :congruence)

Theorem: index-permute-shrink-of-nfix-x

(defthm index-permute-shrink-of-nfix-x
  (equal (index-permute-shrink n count mask (nfix x)
                               numvars)
         (index-permute-shrink n count mask x numvars)))

Theorem: index-permute-shrink-nat-equiv-congruence-on-x

(defthm index-permute-shrink-nat-equiv-congruence-on-x
  (implies
       (nat-equiv x x-equiv)
       (equal (index-permute-shrink n count mask x numvars)
              (index-permute-shrink n count mask x-equiv numvars)))
  :rule-classes :congruence)

Theorem: index-permute-shrink-of-nfix-numvars

(defthm index-permute-shrink-of-nfix-numvars
  (equal (index-permute-shrink n count mask x (nfix numvars))
         (index-permute-shrink n count mask x numvars)))

Theorem: index-permute-shrink-nat-equiv-congruence-on-numvars

(defthm index-permute-shrink-nat-equiv-congruence-on-numvars
  (implies
       (nat-equiv numvars numvars-equiv)
       (equal (index-permute-shrink n count mask x numvars)
              (index-permute-shrink n count mask x numvars-equiv)))
  :rule-classes :congruence)