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Truth

Env-permute-stretch

Signature
(env-permute-stretch n count mask env numvars) → perm-env
Arguments: n — Guard (natp n).; mask — Guard (natp mask).; env — Guard (natp env).; numvars — Guard (natp numvars).
Returns: perm-env — Type (natp perm-env).

Definitions and Theorems

Function: env-permute-stretch

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

Theorem: natp-of-env-permute-stretch

(defthm natp-of-env-permute-stretch
  (b* ((perm-env (env-permute-stretch n count mask env numvars)))
    (natp perm-env))
  :rule-classes :type-prescription)

Theorem: normalize-count-of-env-permute-stretch

(defthm normalize-count-of-env-permute-stretch
 (b* nil
   (implies
        (syntaxp (not (equal count ''nil)))
        (equal (env-permute-stretch n count mask env numvars)
               (let ((count nil))
                 (env-permute-stretch n count mask env numvars))))))

Theorem: lookup-in-env-permute-stretch

(defthm lookup-in-env-permute-stretch
  (b* ((?perm-env (env-permute-stretch n count mask env numvars)))
    (equal (env-lookup k perm-env)
           (env-lookup (index-permute-shrink n nil mask k numvars)
                       env))))

Theorem: env-permute-stretch-of-env-update

(defthm env-permute-stretch-of-env-update
 (b* nil
  (equal
      (env-permute-stretch n count mask (env-update k val env)
                           numvars)
      (env-update (index-permute-stretch n nil mask k numvars)
                  val
                  (env-permute-stretch n count mask env numvars)))))

Theorem: env-permute-stretch-of-nfix-n

(defthm env-permute-stretch-of-nfix-n
  (equal (env-permute-stretch (nfix n)
                              count mask env numvars)
         (env-permute-stretch n count mask env numvars)))

Theorem: env-permute-stretch-nat-equiv-congruence-on-n

(defthm env-permute-stretch-nat-equiv-congruence-on-n
  (implies
       (nat-equiv n n-equiv)
       (equal (env-permute-stretch n count mask env numvars)
              (env-permute-stretch n-equiv count mask env numvars)))
  :rule-classes :congruence)

Theorem: env-permute-stretch-of-nfix-mask

(defthm env-permute-stretch-of-nfix-mask
  (equal (env-permute-stretch n count (nfix mask)
                              env numvars)
         (env-permute-stretch n count mask env numvars)))

Theorem: env-permute-stretch-nat-equiv-congruence-on-mask

(defthm env-permute-stretch-nat-equiv-congruence-on-mask
  (implies
       (nat-equiv mask mask-equiv)
       (equal (env-permute-stretch n count mask env numvars)
              (env-permute-stretch n count mask-equiv env numvars)))
  :rule-classes :congruence)

Theorem: env-permute-stretch-of-nfix-env

(defthm env-permute-stretch-of-nfix-env
  (equal (env-permute-stretch n count mask (nfix env)
                              numvars)
         (env-permute-stretch n count mask env numvars)))

Theorem: env-permute-stretch-nat-equiv-congruence-on-env

(defthm env-permute-stretch-nat-equiv-congruence-on-env
  (implies
       (nat-equiv env env-equiv)
       (equal (env-permute-stretch n count mask env numvars)
              (env-permute-stretch n count mask env-equiv numvars)))
  :rule-classes :congruence)

Theorem: env-permute-stretch-of-nfix-numvars

(defthm env-permute-stretch-of-nfix-numvars
  (equal (env-permute-stretch n count mask env (nfix numvars))
         (env-permute-stretch n count mask env numvars)))

Theorem: env-permute-stretch-nat-equiv-congruence-on-numvars

(defthm env-permute-stretch-nat-equiv-congruence-on-numvars
  (implies
       (nat-equiv numvars numvars-equiv)
       (equal (env-permute-stretch n count mask env numvars)
              (env-permute-stretch n count mask env numvars-equiv)))
  :rule-classes :congruence)