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Truth

Env-permute-shrink

Signature
(env-permute-shrink 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-shrink

(defun
  env-permute-shrink
  (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-shrink))
       (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 (if (eql bit 1)
                     (env-move-var-down count n env)
                     env)))
           (env-permute-shrink (1+ n)
                               (+ bit count)
                               mask env numvars))))

Theorem: natp-of-env-permute-shrink

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

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

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

Theorem: env-permute-shrink-of-env-permute-stretch

(defthm
 env-permute-shrink-of-env-permute-stretch
 (equal
  (env-permute-shrink n count1 mask
                      (env-permute-stretch n count mask env numvars)
                      numvars)
  (nfix env)))

Theorem: env-permute-stretch-of-env-permute-shrink

(defthm
 env-permute-stretch-of-env-permute-shrink
 (equal
  (env-permute-stretch n count1 mask
                       (env-permute-shrink n count mask env numvars)
                       numvars)
  (nfix env)))

Theorem: lookup-in-env-permute-shrink

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

Theorem: env-permute-shrink-of-env-update

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

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

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

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

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

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

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

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

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

Theorem: env-permute-shrink-of-nfix-env

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

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

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

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

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

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

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