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• Truth

Env-permute-polarity

Signature

(env-permute-polarity n mask env numvars) → new-env

Arguments

n — Guard (natp n).

mask — Guard (integerp mask).

env — Guard (natp env).

numvars — Guard (natp numvars).

Returns

new-env — Type (natp new-env).

Definitions and Theorems

Function: env-permute-polarity

(defun env-permute-polarity
       (n mask env numvars)
       (declare (xargs :guard (and (natp n)
                                   (integerp mask)
                                   (natp env)
                                   (natp numvars))))
       (declare (xargs :guard (<= n numvars)))
       (let ((__function__ 'env-permute-polarity))
            (declare (ignorable __function__))
            (b* (((when (mbe :logic (zp (- (nfix numvars) (nfix n)))
                             :exec (eql n numvars)))
                  (lnfix env))
                 (env (if (logbitp n mask)
                          (env-swap-polarity n env)
                          env)))
                (env-permute-polarity (1+ (lnfix n))
                                      mask env numvars))))

Theorem: natp-of-env-permute-polarity

(defthm natp-of-env-permute-polarity
        (b* ((new-env (env-permute-polarity n mask env numvars)))
            (natp new-env))
        :rule-classes :type-prescription)

Theorem: lookup-in-env-permute-polarity-split

(defthm lookup-in-env-permute-polarity-split
        (b* ((?new-env (env-permute-polarity n mask env numvars)))
            (equal (env-lookup m new-env)
                   (if (and (<= (nfix n) (nfix m))
                            (< (nfix m) (nfix numvars)))
                       (xor (env-lookup m env)
                            (logbitp m mask))
                       (env-lookup m env)))))

Theorem: lookup-in-env-permute-polarity-set

(defthm
 lookup-in-env-permute-polarity-set
 (b*
  nil
  (implies
       (and (logbitp m mask)
            (< (nfix m) (nfix numvars)))
       (equal (env-lookup m
                          (env-permute-polarity 0 mask env numvars))
              (not (env-lookup m env))))))

Theorem: lookup-in-env-permute-polarity-unset

(defthm
 lookup-in-env-permute-polarity-unset
 (b*
  nil
  (implies
       (and (not (logbitp m mask))
            (< (nfix m) (nfix numvars)))
       (equal (env-lookup m
                          (env-permute-polarity n mask env numvars))
              (env-lookup m env)))))

Theorem: env-permute-polarity-of-swap-polarity

(defthm
 env-permute-polarity-of-swap-polarity
 (b*
  nil
  (equal
    (env-permute-polarity n mask (env-swap-polarity m env)
                          numvars)
    (env-swap-polarity m
                       (env-permute-polarity n mask env numvars)))))

Theorem: env-permute-polarity-reverse

(defthm env-permute-polarity-reverse
        (b* ((?new-env (env-permute-polarity n mask env numvars)))
            (equal (env-permute-polarity n mask new-env numvars)
                   (nfix env))))

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

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

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

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

Theorem: env-permute-polarity-of-ifix-mask

(defthm env-permute-polarity-of-ifix-mask
        (equal (env-permute-polarity n (ifix mask)
                                     env numvars)
               (env-permute-polarity n mask env numvars)))

Theorem: env-permute-polarity-int-equiv-congruence-on-mask

(defthm
   env-permute-polarity-int-equiv-congruence-on-mask
   (implies (int-equiv mask mask-equiv)
            (equal (env-permute-polarity n mask env numvars)
                   (env-permute-polarity n mask-equiv env numvars)))
   :rule-classes :congruence)

Theorem: env-permute-polarity-of-nfix-env

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

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

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

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

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

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

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